The answer is probably not as definitive as you'd like. By some academic analyses (almost consensus, one could say)
"women had no status in the Constitution of 1787"
While the constitution used some gender-neutral language, early interpenetration in that direction e.g. voting rights for women in New Jersey were reversed by 1807. (This a bit O/T, but the topic has generated two comments, so here's what the linked paper [authored by a history prof] says about that:
Yet not all male New Jerseyites greeted woman suffrage with such
exuberant glee. [...] In 1807, relying on the persistence of such attitudes, John
Condict took his revenge for his near-defeat by women ten years
earlier: he introduced the bill that successfully disfranchised both
women and property-owning free blacks, arguing that the votes of
such persons were more likely to be corrupted than were those of
independent white males.
)
That view is echoed in a longer piece of (Supreme) justice Sandra Day O'Connor, who attributes the practical striking down of laws discriminating against women to the post-1970s reinterpretation of 14th Amendment by the Supreme Court. More officially, these should have been enshrined in the Equal Rights Amendment (for women) of 1972, but that amendment never came to pass, although its failure spurred many concrete substitutes in narrower areas:
Women emerged in significant numbers all across the country in the 1960s to demand equal
opportunity, primarily in the workforce. Pursuant to its power under the
Commerce Clause, Congress enacted both the Equal Pay Act of 1963 and Title Vll
of the Civil Rights Act of 1964 prohibiting employment discrimination on the basis
of race or sex. In 1972, Congress sent the proposed Equal Rights Amendment to
the states for ratification. In response, although it was not ratified, many states
became active in reviewing state legislation to remove discriminatory laws and to
pass state civil rights legislation.
The Supreme Court began to look more closely at legislation providing
dissimilar treatment for similarly situated women and men in the early 1970s. The
first case in which the Court found a state law discriminating against women to be
unconstitutional was Reed v. Reed. The case was decided in 1971, more than 100
years after the ratification of the 14th Amendment. Applying only a rationality
standard, the Court struck down an Idaho law giving men an automatic preference
in appointments as administrators of estates. Following Reed, the Court invalidated
a broad range of discriminatory statutes under the Equal Protection Clause of the
14th Amendment. For example, a federal law providing for determination of a
spouse's dependency based on the sex of the member of the armed forces claiming
the benefits; a Social Security Act provision allowing widows bur not widowers to
collect survivors benefits; state law requiring divorced fathers to support their
sons until age 21 but their daughters only to age 18; a state law permitting the sale
of beer to women at age 18 but not to men until age 21; a state law requiring men
but not women to pay alimony after divorce; and a state statute granting only
husbands the rights to manage and dispose of jointly owned property without the
spouse's consent. In 1976, in the case of Craig v. Boren, the Court adopted a
somewhat stricter standard of review for sex, based classifications and held that to
"withstand constitutional challenge [under the Equal Protection Clause] ...
classifications by gender must serve important governmental objectives and must be
substantially related to achievement of those objectives."
All in all, the Court has heard over 50 cases since 1971 involving various sex,
based challenges under the Equal Protection Clause to stare and federal laws relating
to hiring, promotions, maternity leave, disability insurance, pension rights and
seniority. Some of the challenges have been brought by women, some by men. Not
all such challenges have been successful. But there is no question that the Court has
now made clear that it will no longer view as benign those archaic and stereotypic
notions concerning the roles and abilities of males and females. A statute classifying
people on the basis of sex will not be upheld without an exceedingly persuasive
justificanon for the classification.
So given the principles enunciated in Boren (1976), it's probably fair to say that challenging women's right to be elected would probably fail miserably in front of the Supreme Court thereafter, in view all the other (50 or so) laws struck down.
Having said this, a 1976 article notes that while the 14th Amendment Equal Protection is the most likely legal safeguard of the right to be elected, there are some alternative (legal) views some tying the candidacy with the right to vote:
Protection of the right to vote through the Fourteenth Amendment has been the
most frequent justification for invalidation of restrictions on candidacy. The
right to vote, and, more importantly, the right to an effective vote, has served as
the basis for application of a compelling state interest standard to any regulation
impinging upon that right. Protection of candidates' rights is traced
through a circuitous route whereby it is shown that voters' rights are abridged
by the impairment of a candidate's ability to gain access to the ballot. The right
to vote, it is argued, loses its importance in the absence of a meaningful choice for
the voter, and hence restrictions on candidates' opportunities to run for office
indirectly restrict voters as well.
This theory, taken to its purely logical conclusion, would require that any
infringement on candidacy be viewed as an infringement upon voters' rights,
but this approach does not seem to have been adopted anywhere.
Williams v. Rhodes provided the first indication at the Supreme Court level
that voters' rights could be impaired through candidacy restrictions. The Court,
per Justice Black, announced that the Ohio laws in question placed heavy burdens,
not only on the right to associate, but also on "the right of qualified voters,
regardless of their political persuasion, to cast their votes effectively." While
acknowledging that "the State is left with broad powers to regulate voting," the
Court rejected arguments that Ohio had a compelling interest in promoting the
two-party system, in seeing that the election winner be the choice of a majority
of the voters, and in preventing voter confusion due to the remote possibility
of the presence of too large a number of parties on the ballot.
Bullock v. Carter, a 7-0 decision, reinforced the identification of voters'
rights with candidates' rights. The statute there in question required filing fees
for local office in Texas primary elections as high as $8,900 without any writein
vote or other alternative method by which candidates could be entered on the
ballot. Chief Justice Burger's opinion for the Court discussed the law in terms
of the rights of voters:
The initial and direct impact of filing fees is felt by aspirants for office, rather
than voters, and the Court has not heretofore attached such fundamental status to
candidacy as to invoke a rigorous standard of review. However, the rights of voters
and the rights of candidates do not lend themselves to neat separation; laws that
affect candidates always have at least some theoretical, correlative effect on voters.
However none of these (latter) cases involved sex-based discrimination, so the connection is more tenuous/theoretical. They also largely pre-dated most of the other ones in which the Supreme court drew boundaries on sex-based discrimination. So, saying that the 19th Amendment has some implications on women's right to candidacy might also have a legal leg to stand on.
And since two answers here rely on the interpretation of pronouns in the Constitution... that's also an argument but not an incredibly convincing argument, alas. The election of the first woman to Congress did precede the 19th amendment, but not by much... and a law review around that time pointed out that courts tended interpret the ambiguity against women.
III. HE/HIM/HIS
All pronoun references to the President, the Vice President, Senators and
Representatives, and other officers are masculine. Some version of a male pronoun
appears close to 50 times in the Constitution. Indeed, the qualifications for electors
in the Fourteenth Amendment is specifically stated as “male.”
Use of the male pronoun to refer to all humans, according to linguist Dennis
Baron, can be traced back as far as the Latinists of the sixteenth century and was
widely accepted in the eighteenth century. Women seeking voting rights argued
that, if the male pronoun was general to all sexes, its exclusive use in suffrage
statutes could not pose any impediment to the suffrage of women. Equally, they
argued, it could not bar women from being able to take up elected office. A note
in the Harvard Law Review in 1910 summed up the chauvinism of the time, stating
that “although the exclusive use of masculine pronouns in the constitutions in this
country has never been regarded as excluding women, there has been little
tendency to construe general provisions in their favors.” The article went on to
observe that contemporary courts had tended to construe any ambiguity against
female office-holding.
That attitude did not seem to deter Sara Platt Decker of Denver, Colorado,
however, who considered a run for Congress in 1909. Speculation about a female
congressional candidate sparked one opinion writer to object — “[s]trict adherents
to the letter of the Constitution maintain that the presence of the masculine
pronoun, and the absence of any other, obviously renders ineligible any person of
the feminine persuasion.” It seems, however, that this writer held a minority
view. Jeanette Rankin, of Montana, became the first elected female congress
member in 1916 – four years before ratification of the 19th Amendment. And,
although there was a bit of grumbling by sources to the Washington Post about
pronouns and Montana’s new representative, her swearing in took place without
much pronoun based objection. By 1922, the idea that she could have been barred
from office because of pronouns barely merited a sentence in Joseph Ragland
Long’s treatise on American Government: “[T]he pronoun ‘he’ [in Article I]
includes both sexes.”
Today, the assumption that “he” means “he or she” has become so
entrenched, that when former Senator and Secretary of State Hillary Clinton ran
for President in 2015, no one in mainstream legal circles attempted to argue that
she was ineligible for the Presidency.
Although these case have been basically forgotten by now, the original 1910 article (Note, Eligibility of Women for Public Office, 24 HARV. L. REV. 139, 140 (1910)) provides some examples in which the lack of voting rights was interpreted as extending to lack eligibility:
Most constitutions
restrict suffrage to males, and even where eligibility for office
is not expressly confined to electors, it would seem naturally to be
predicated on the right to exercise this primary governmental function.
On this ground, several cases have denied women the right to hold
office.
[footnote:] See Atty.-Gen. v. Abbott, supra; Atchison v. Lucas, supra. But see State v. Hostetter,
supra; Wright v. Noell, supra. It has been said that conferring suffrage on women
makes them eligible for office. See State v. Cones, I5 Neb. 444. Cf. Olive v. Ingram,
2 Strange III4. But in England it has been held that a woman is not eligible even
for an office for which she can vote. Beresford-Hope v. Lady Sandhurst, supra.
So the all-inclusive male-pronoun was hardly that convincing before the 19th Amendment.
