No. Properly, the word ‘gay’ is used as an adjective – i.e., ‘a gay man’ – and not as a noun – i.e., ‘a gay’.
The word ‘gay’ used to mean ‘full of joy’, ‘merry’, ‘happy’. Over the course of the twentieth century, however, it came to be used to mean ‘homosexual’, and nowadays, this is the only meaning most people associate with the word. (If you read older texts, you are likely to see the word used with its older meaning.)
The word ‘gay’ is usually used as an adjective – i.e., ‘a gay man’. Of course, there’s nothing stopping anyone from using any word how they like, and you could use the word ‘gay’ as a noun – i.e., ‘a gay’. There are many people who do so – indeed it’s a particularly common speech pattern among Americans.
However, using the word ‘gay’ as a noun tends to sound jarring – in the same way that referring to ‘a tall man’ as just ‘a tall’ would. In order to sound fluent, it is better to use ‘gay’ only as an adjective.
The word ‘gay’ can be used to refer to men or women – ‘a gay man’ is a homosexual man, and ‘a gay woman’ is a homosexual woman – but it is uncommon to use it for women. It’s more common to use the word ‘lesbian’ for women. (The word use is not perfectly symmetric, and the word ‘lesbian’ can be used as either an adjective or a noun – i.e., ‘a lesbian woman’ or ‘a lesbian’.)