Are they any current openly gay Major League Baseball players?
The answer is no. In fact, I’m pretty sure none were openly gay during their careers. A few have come out after their careers. But not really the same.
Baseball seems to be doing better and is probably ready for an openly gay player, but being the first will be tough. If they were fortunate, a small group could come out at the same time.
Or a couple. 
Besides Carl Nassib in the NFL, I’m having trouble thinking of anyone who’s come out in the major sports.
That’s a different story. If two players find love or even just have a physical attraction, more power to you, I’m happy for you. But I imagine if players get romantically involved that might make things awkward, especially as players get traded and have to separate. Imagine if they are competing against each other in a game! “Sorry I hit you with that pitch honey!”
Then again, maybe it wouldn’t be a big deal. I think that would be interesting to see.
Jason Collins, NBA player (now retired):
Of course, in women’s sports, it’s common.
Bad idea for any business. Most romance ends in break up and break ups bring bitter feelings. In most business one partner can quit. Thats not possible in professional sport where players play under contract. And the same harassment rules work for both traditional and same sex couples so messing around with non players can be a minefield too.
Depends on what you consider a major sport, I guess. In MLS, former player Robbie Rodgers came out as gay in 2013 - technically he was ‘retired’ at the time, but the Galaxy signed him after. And Colin Martin of Minnesota United is openly gay.
How soon they forget. Michael Sam was the 2013 SEC Defensive Player of the Year at Missouri. The Rams drafted him in the seventh round, which got a lot of attention, but he got cut in training camp. He did go on to play one year with the Montreal Alouettes of the CFL.
Yeah, I’d say MLS counts.
19 year old NHL prospect Luke Prokop is openly gay. He hasn’t played in the NHL yet but likely will this upcoming season.
There are a fair number of openly gay female athletes in the WNBA, and on the U.S. Women’s National (Soccer) Team. That said, women’s sports in the U.S. seem to have been more accepting of homosexuality than men’s sports have.
So, as far as the original question is concerned, the answer is still zero?
I believe that’s correct.
This 2020 article indicates that, as @What_Exit noted earlier, there’s been very few (in fact, only two) MLB players who came out as gay, and both only did so after retiring: Glenn Burke, who played in the '70s, and Billy Bean, who played in the '80s and '90s. Burke was apparently open with his teammates, though he was publicly closeted during his playing career.
My first cousin once removed was a closeted gay MLB umpire in the '80s. Does that count for anything?