In short: No.
He has written some emotional, sometimes called homo-erotic poems to a dear friend ( Tommaso dei Cavalieri) , and due to this fact and his David, who sadly became an idol of the gay movement, he is sometimes seen as gay, BUT:
-writing poems like this was very common as a form of antiquity fever in the renaissance. Expressing love for a male friend like this is not always a sign for real sexual contacts, but for a sensitive mind, a quality highly in favor for the noble elite.
-there are stories about his chaste life. There is no more security in this, too, but we do have an opposite meaning about him.
He is described as a devout catholic, too.