It is well known that the derivatives of Latin mulier and fēmina competed in Romance languages as the main word for `woman'. For instance, the former remained as Spanish mujer and Portuguese mulher (though S hembra, P fêmea do remain). French language, however, has, to my knowledge, only femme.
The question is: does any descendant of Latin mulier remain in French language and if not, what would the gistorical derivative form look like if it survived?
The Trésor de la langue française has most the answer to your question in the etymology section for femme:
From Classical Latin femina “female”, then “woman, wife” which competed against the Latin words mulier “woman” which no longer survives in French (contrast with Italian moglie, Spanish mujer) except in the archaic form moillier “wife, woman” (which disappears from texts in the 14th century; 2 occurrences in the 15th and 16th centuries), and uxor “wife” which gave the very rare oissour “wife” (which disappears from texts in the first half of the 13th century; a few occurrences remain in the 13th and 14th centuries in epic retellings).
So, no, it didn't survive (nor did any word based on this root that I can think of or find in a word list). The Middle French form would be moillier, which might have evolved to *mouiller or *mouille (parallel to moiller → mouiller”) or possibly to something simpler if it had been a common word (possibly via variants such as muiller or molier which could have led to *molie or *mole).
Oddly, there is a rare modern French word which has the same root, through a different path: moukère or mouquère, which means a woman, from the Algerian Arabic word moukera which was a borrowing from the Spanish mujer.
