Question: Does Amy Coney Barrett’s record suggest she would vote to overturn Obergefell v. Hodges, and what specifically? Or is this just an assumption based on her right-leaning stance on other issues?
Unclear, but likely. Unclear because we have no record of Judge Barrett speaking on either Gay Marriage generally or the Obergefell v Hodges case specifically. Likely because Judge Barrett has been nominated by a President and party who’s stated that repeal of Obergefell v. Hodges* is a top priority and even litmus test for its Supreme Court Nominees.
The GOP’s 2020 platform “condemns” marriage equality.
2020 GOP Platform under section "Defending Marriage against Judicial Activism": “Our laws and our government’s regulations should recognize marriage as the union of one man and one woman … We do not accept the Supreme Court’s redefinition of marriage and we urge its reversal.”
Likewise Conservatives on the court are already telegraphing their intentions with regards to their new super majority.
Obergefell v. Hodges was a 5-4 decision. Giving conservative judges a new 6-3 majority could endanger Obergefell v. Hodges (gay marriage) which became national law, not due to legislative action, but on constitutional grounds after California passed (Assembly Bill 19 (AB 19)) legalizing Gay Marriage and then repealed it ( Prop 8 2008i). Barrett's nomination could, but also may not, endanger gay marriage and Obergefell v. Hodges directly.
One of the first court cases a new Judge Barrett will hear is Fulton v. Philadelphia; (days after the presidential election) a case which could set new precedent for conflicts between religious practices and governmental nondiscrimination rules; thus weakening minority rights in favor of religious beliefs.
Still Barrett's position, going either way, is not a slam dunk. As a strong Catholic who attended Catholic Hight school, College and Graduate Schools one might think she would side with conservative religious position on these cases. On the other hand, her named mentor on the Supreme Court was Anthony Scalia who voted with the majority originally that the State's ability to legislate over religions was not a violation of the First Amendment, if the legislation in question applied broadly and did not single out a specific religion (Employment Division v. Smith in 1990). Still Employment Division vs Smith involved a native American church's right to receive unemployment benefits while smoking peyote as a religious practice. Scalia's vote would have been more out of character agreeing with Mr. Smith than voting with the majority and denying him his employment checks and curbing the first amendment.