Was Bugs Bunny a gay character?

No.

Despite some news reports about Steve Blanc, alleged son of voice actor Mel Blanc, saying Bugs Bunny is gay and What's up, Doc? is a passphrase for a gay nightclub, it is just a joke article by Citizen of the Month.

Note, November 2010: This is not true. It is a joke. This was NOT approved by Warner Brothers. And Mel’s son’s real name is Noel. And just so you know — Bugs Bunny is probably my biggest literary influence.

Mel's son isn't called Steve and the voice actor does not decide the sexuality of the character, the studio (Warner Brothers) does.

Thanks to Paulie_D for the reminder, Bugs Bunny also had a girlfriend, Lola. This seems to make it even more likely that he was not a homosexual.

I would wager that there cannot be any definitive proof, unless the studio actually states something like He is not gay., but I find it very unlikely that he would be a closet homosexual in a fake relationship and in-love with a male from another race.

There is no evidence Bugs is gay and plenty of evidence to the contrary.

The sexual orientation of a fictional character is completely up to those who have creative control over the character. Since most of those people who created Bugs Bunny are now deceased having never claimed that the character was meant to be gay, a conclusive answer is not likely to ever be available. However, there is a great deal of evidence of what those creators intended for the character's sexual orientation and it overwhelmingly points toward heterosexual.

Whenever Bugs Bunny is shown to have a genuine sexual interest, it is always with a female character (or with a decoy-type object that he perceives to be a female character). Some examples:

Cut to the last 10-15 seconds of this clip to see Bugs' encounter with Lauren Bacall...

Bugs was fooled (and attracted) by mechanical female rabbits on two occasions...

In Bewitched Bunny, the character Witch Hazel transforms into a female rabbit at the end and wins Bugs' affection.

On the other hand, Bugs' cross dressing was always done to fool an antagonist and kisses planted on the faces of male characters were always mocking and taunting, never depicted as a product of Bugs' desire.

While it is perhaps tempting for some people to try to make the case that a popular character from an era in which homosexuals were not represented in popular culture was secretly gay somehow, it is just wishful thinking. The overwhelming evidence points to the character being a heterosexual.