My recollection differs from this, but that could be my memory of regional (deep south) politics rather than national politics. Republicans on the state level definitely reacted to Lawrence v Texas (2003) and gay marriage becoming legal in Massachusetts, which technically started in 2004 but the court case was won in 2003 as well. And it’s not like gay marriage wasn’t a right wing boogeyman for like a decade before that as well. But yeah, it got turbo charged during the end of the first Bush term.
Additional info for anyone that isn’t also an old, ‘came out swinging against gay marriage’ means he was campaigning on making a constitutional amendment to ban gay marriage. Thank fuck that went nowhere, at least.