The answer is probably not, but to answer with certainty that would require a Supreme Court decision.
There are two clauses of the Constitution relevant here. The first is Article 4's full faith and credit clause, which requires states to recognize the "public acts, records, and judicial proceedings" of any other state. The courts have not yet ruled definitively about whether this requires states to recognize other states' marriages. Courts have arrived at different decisions regarding the Full Faith and Credit clause's applicability to marriage licenses and family law, but if a state did recognize another state's marriage license, the issue would be irrelevant. A state that recognized another state's marriage license could not face Constitutional challenges on the Full Faith and Credit Clause.
However, it could face challenges on Fourteenth Amendment grounds. The 14th Amendment requires that states guarantee their citizens equal protection under the law. Laws discriminating against homosexual marriage would likely be ruled unconstitutional on the grounds that they violate the guarantee of equal protection.
Though the Supreme Court has not ruled regarding state laws regarding homosexual marriage, the Supreme Court did rule in United States v. Windsor that the Federal government had to recognize same sex marriages (though it did not require that states hand out same sex marriage licenses) because the Fifth Amendment guarantees equal protection under the law. The Fifth Amendment necessitates that the Federal government provide citizens equal protection under the law, and the Supreme Court found that that prevented the federal government from discriminating against homosexual marriages. The Fourteenth Amendment necessitates that states provide citizens equal protection under the law, so the same logic makes it unconstitutional for states to refuse to recognize homosexual marriages.
That's not just my logic. Though the Supreme Court has yet to rule on the issue of whether state governments can refuse to recognize homosexual marriages, many lower courts have. Judges in Utah and Oklahoma have recently ruled those states' bans on same sex marriage unconstitutional on Fourteenth Amendment grounds. The same logic was also used to overturn bans on same sex marriage in New Jersey, California, and elsewhere. So we can reasonably conclude that, though your proposed law would not face challenges on Full Faith and Credit grounds, it would on Fourteenth Amendment grounds.