Is ‘smart alec (aleck)’ a gender-neutral noun?

I happened to find the origin of the word, ‘smart alec (aleck / alick)’ as I was checking the spelling of the word in a dictionary.

Merriam Webster English Dictionary defines ‘smart alec (aleck / alick)’ as:

An obnoxiously conceited and self-assertive person with pretensions to smartness or cleverness.

And its origin as;

Aleck, nickname for Alexander

First Known Use: 1865.

Now that knowing ‘smart alec’ being a shortened form of ‘smart Alexander,’ I became a bit ill at ease, because I’ve used the word for both sex thus far without any reflection on its origin and gender.

Is there a female counterpart to smart alec? Is it alright to use ‘smart alec (aleck / alick)’ as usual for both a man and a woman?

Yes, it can be used for either gender. A Smart Alec is a neutral noun - there is no gender attached or implied.

Additionally, as @TrevorD commented, Alec, Alex, and other similar contractions are used for female names as well as for male names.

Similarly, while the root of 'Billy No-Mates' is male, it can be used to describe an unpopular man or woman.