That's actually a myth.

[...] in England, where parliament had made homosexuality a capital offense in 1533, hanging was the method prescribed. Use of faggot in connection with public executions had long been obscure English historical trivia by the time the word began to be used for "male homosexual" in 20th century American slang, whereas the contemptuous slang word for "woman" (in common with the other possible sources or influences listed here) was in active use early 20c. [...]

You can read more under the second noun listing here:

https://www.etymonline.com/word/faggot