The specific provision would be Article IV, Section 1 of the US Constitution which specifies that US states must recognize "public acts, records, and judicial proceedings of every other state."
http://en.wikipedia.org/wiki/Full_Faith_and_Credit_Clause
There has yet to be a significant court ruling on whether the Federal Defense of Marriage Act violates this section of the constitution if marriages are considered part of this provision. Such a ruling would come close to defacto legalizing gay marriage everywhere in the US since gay couples would merely have to travel to a state where gay marriage is legal and they would suddenly be legally married no matter where they live.
As I noted though, with no cases currently pending on these grounds making their way through the court system, it likely would be awhile before the Supreme Court could possibly come down with such a ruling. (Although a ruling outright stating a government bans on gay marriage are unconstitutional could potentially happen more quickly.)