Wearing my professional historian’s hat, I’d have to say he wasn’t gay, because ‘gay’ wasn’t an identity in Shakespeare’s time. In the fifteenth and sixteenth centuries, there was an admiration for the beauty of women and for boys, in terms of androgyny. Christopher Marlowe wrote about it as well as Shakespeare, namely older men looking at younger people, which makes us incredibly uncomfortable today as it goes against our ways of looking at sexual desire. If you read a sixteenth century love poem to a lady, the writer was most probably sexually attracted to that lady, so the language that Shakespeare at that time meant he was showing an attraction to both women and boys: he’s not in the non-sexual realm of male-to-male social bonding, it’s clearly erotic desire.
"Interestingly, for most of history the queerness of Shakespeare was erased, because the sonnets were mostly published in altered editions"
So, in modern terms, you could say Shakespeare was bisexual with a bit of paedophilia stirred in, which was pretty normal and perfectly respectable in those days. When I say normal, there is an argument in medieval and Arabic society that the traditional pattern was for men to be attracted to women, and to boys, but not to other adult men, but then when the boy hits puberty, and starts growing a beard, the man loses interest in the boy.
Rufus Wainwright performs Sonnet 20, which contains ambiguous references to beauty
It was the same in Ancient Greece and Rome. But that pattern changed in the eighteenth century, when same-sex desire was thought to be wrong in all forms. I think it really does matter whether Shakespeare was gay or not, because he is used as a yardstick of the value of English culture, so it makes a huge difference if you can say that the greatest writer in the English language is actually not classified as ‘normal’. Interestingly, for most of history, the queerness of Shakespeare was erased, because the sonnets were mostly published in altered editions. It was only later, in the nineteenth century, that people discovered that they weren’t reading the original texts, when they went back to the original text.