Whether "gasket" comes from French "garcette" or not, I have no idea.
If a particularly misogynistic and at the same time naval etymology is sought, then the thing called "garcette" was, among other uses, an instrument of corporal punishment.
However, that's probably not the original meaning in French.
This source claims that it came from Spanish "garceta" and was reportedly inspired by the 17th century hairstyle of this lady.
That hairstyle is supposed to resemble a heron's ("garza" in Spanish), or perhaps an egret's ("garceta" in Spanish) feathers. In another meaning, "garceta" refers to a certain hairstyle in Spanish even today.
As a loanword in French it has quickly undergone a reanalysis including adaptation of the spelling.
If this Spanish to French theory is correct, English can hardly borrowed the word exclusively from French - the timeline does not match.
This former pirate has used the word in English (in Voiage into the South Sea) already in 1622 or, I suppose, several years earlier. I don't own the book and I'm not sure of the spelling he or his editors chose there. In contrast, Anne of Austria became Queen Consort of France only in 1615 (an important milestone for the relevance of her hairstyle in France) and "garcette" has the first known record in French in the naval meaning, initially still spelt "garsette", only as of 1636.
Of course, subsequent influence of French on English is entirely possible. Distance sailing is about language contact, right?
(To minimize any additional confusion about whether there ever exists any Spanish word spelt as "garcette" or whether any edition of the Century Dictionary mentions such a French-sounding spelling in Spanish: There never was such a word in Spanish. The Century Dictionary probably never claimed its existence either. The web page quoted in the OP is at fault for this inaccuracy.)