Lieberman’s position, when running for the presidency in 2004, was basically: “No same-sex marriage anywhere; there can be civil unions in states that want them, but those rights should be doled out to same-sex couples on a right-by-right basis.”
He was trying too hard to “be different” and pander to social conservatives, when, arguably, all of the rest of the Democratic candidates in the presidential field that year each had a more progressive stance on civil unions (even when very few of them would support full same-sex marriage itself).
I don’t know if Joe Lieberman has changed his position, in the nine years since he’s retired from Congress; but, even if he has, he’d have nothing to lose in the present. Hell, he would have had nothing to lose by supporting gay marriage as a U.S. Senator from Connecticut when he WAS in office.