Does the word fascist share a root with the anti-gay slur f****t?
Both terms refer to a bundle of sticks, but in the case of fascist it refers to the Ancient Roman usage as a symbol of power, while in the case of faggots it appears to refer to a worthless bundle, originally used as a derogatory term to refer to a woman. Other suggested origins don’t seem to have any relation to the “fascist” usage:
Fasces:
1590s, from Latin fasces "bundle of rods containing an axe with the blade projecting" (plural of fascis "bundle" of wood, etc.
Carried before a lictor, a superior Roman magistrate, as a symbol of power over life and limb: the sticks symbolized punishment by whipping, the axe-head execution by beheading. Hence in Latin it also meant, figuratively, "high office, supreme power."
Faggot:
The explanation that male homosexuals were called faggots because they were burned at the stake as punishment is an etymological urban legend.
Burning sometimes was a punishment meted out to homosexuals in Christian Europe (on the suggestion of the Biblical fate of Sodom and Gomorrah), but 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., by D.H. Lawrence and James Joyce, among others.
(Etymonline)
In addition to faggot, a known decedent of the Latin fasces, a bundle of sticks, there may be another faggot in American English that became the pejorative for male homosexual.
פֿייגעלע (feygele) , Yiddish for little bird may actually be the source of the pejorative. Feygele was used in early 20th Century New York the same way faggot is used today.
Faggot in its current pejorative sense seems to have originated in early 20th Century America. This would have been the same time period that saw feygele in American Yiddish in the pejorative sense.
There is no point trying to prove that Yiddish slang has contributed to American slang; it has beyond question. And no point trying to prove that feygele was the sole origin of the politically incorrect faggot. There are endless possibilities here. Among them is that faggot was the closest sounding English word to feygele some Americans could come up with.
Tracing faggot to the German vogel rather than the Latin fasces would have seemed a stretch to me some years ago. But, I think the origin of this term has not been sufficiently examined. Some uses of (fasces) faggot make sense, such as cigarette, as they were once sold in bundles. But that faggot in the sense of a male homosexual, does not.
I am prepared to keep my mind open here.
First, let's look at the basic etymology of the two words.
Faggot etymology:
Middle English (in the sense ‘bundle of sticks for fuel’): from Old French fagot, from Italian fagotto, based on Greek phakelos ‘bundle’.
Oxford Dictionaries
It's important to note that "phakelos" is "φάκελος".
Fascism etymology:
1920s: from Italian fascismo, from fascio ‘bundle, political group’, from Latin fascis (see fasces).
Oxford Dictionaries
According to the Etymological Dictionary of Latin and the other Italic Languages, fascis is not related to phakelos/φάκελος:
facis 'bundle, faggot; (symbol of) power' [m i] (Pl.+)
IE cognates: Olr. basc 'necklace', W. beich 'burden', Bret. bec'h 'load' < PCl. *b(h)askio-.An Italo-Celtic correspondence *bhaski(o)- 'bundle' for which no PIE etymology is available. Greek words such as φάκελος 'bundle' and βάσκιοι 'bundles of osiers' (Hsch.) are probably unrelated, since they show irregular alternations within Greek, and do not have exactly the same preform *bhaski-(o-).