Sexual orientation won't matter to President Obama -- this I do believe, based on several years of reporting on the guy. Unless he's influenced by subconscious patter, he's not going to choose someone because she's gay, and he's not going to remove someone from a list because she's gay.
That does not mean, in any way, that he won't want to think through the ramifications of what the appointment of the first openly gay jurists would entail. Obama likes to ask his counsels about precedent and history and second-order effects. He will leave the deliberations to the constellation of officials who are participating in the selection process, including his voracious chief of staff, Rahm Emanuel, and Michael Strautmanis, a long-time Obama counselor who works for Valerie Jarrett.
But if Obama makes the choice, what ought to happen will yield to the reality of politics, the structural bias of the news media and the sociology of Congress.
Absent another storyline -- the FIRST HISPANIC, a MAJOR ideological shift on the court -- if Obama selects a gay person, the gay thing becomes the second thing that most people will know about the nominee. (The first, as pointed out by a White House official, is that this person has Obama's imprimatur. This didn't work for an unpopular President Bush, but it might work with Obama.)
The Obama White House has studiously avoided engaging with Republicans on culture war tropes. Believe it or not, it's one of the main reasons why the White House hasn't done more in public for Dawn Johnsen. Given that Rahm Emanuel is running the vetting process, I can't imagine that the White House won't include the upside/downside potential for this type of a rhetorical battle in their calculations and recommendations.
Obama might ignore this, but the White House won't. From their lips to the ears of anyone who asks: Rahm and David Axelrod don't want the GOP base riled up in 2010.
In the end, the nomination of a gay Supreme Court Justice might be the perfect way to engage the public, and I think the upside potential is greater than the downside risk. What better way to advance the cause of the public normalization of homosexuality than to appoint a gay justice to the court without referencing her sexuality?
If, on the other hand, the White House nominates a gay justice and makes a big deal about it and then becomes defensive when social conservatives object, they've undermined their underlying social engineering.
Journalists will cover the issue reductively, interest groups on all sides won't be able to resist; opponents may well use her sexuality as a weapon against her, and proponents will see every attack against her as motivated by antipathy to homosexuality.
In any event, almost every political actor involved in this decision will have to confront the question of gays and society.
Not too long ago, being (openly) gay was enough to disqualify a Clinton appointee, James Hormel, from clearing Congress to be the ambassador to Luxembourg. Clinton used a recess appointment. George Bush's openly gay appointee was given the AIDS portfolio -- not insignificant, but not outside the gay Republican ghetto either.
A side note, as pointed out by someone smart: single, accomplished, older women are automatically assumed to be lesbians
-- witness the demonization of Janet Reno, Janet Napolitano and others -- while openly gay female professionals deal with stigma directly. Is society ready to be gay blind?