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Population:

• male is 100


• female is 90

The population of males is higher than their "gender counterpart"

Or

The population of males is higher than their "female counterpart"

Let's say you are trying your best to find the synonym of "female", so you used the bold words above.
Which of the two is the correct English words that is equivalent to the word female?

I believe neither of those two sentences uses the word "counterpart" correctly. The word "counterpart" does not refer to opposites. Let me give you an example. "The generals met with their counterparts from the enemy army to discuss terms of surrender." In other words, the generals are meeting with the enemy generals. They are alike instead of opposite. They are doing the same kind of job. Here's another example. The President of the United States, Donald Trump, met with his counterpart, the Chancellor of Germany, Angela Merkel. They both hold a similar job, so they are counterparts, even though one is male and one is female.

It would be more correct to write:

The male population is higher than that of the opposite gender.

Or just:

The male population is higher than the female population.

One of the definitions of "counterpart" on www.merriam-webster.com/dictionary says:

Something that completes : COMPLEMENT
"The lead actress and her male counterpart"

Gender counterpart would generally mean the other gender. So if it's for males, the counterpart would be understood to be female and vice versa. (It works here because there are only two genders. If talking about more, this could create confusion.)

Female counterpart would specifically mean the females.

If you're really desperate not to use female or women, you could use the everyday terms opposite sex or (less commonly) opposite gender. If you really want to sound fancy - which is what it sounds like you're trying to do - you could say gender complement, but this is not a widely used term (though it's not completely unattested). It just makes sense in terms of what complement means¹, at least if you take a heteronormative view (that is to say, the idea that a man and a woman 'complete' one another). Counterpart isn't a natural thing to use in this situation at all.

Really, you're better off with a simpler and less obscure term.

¹: Interestingly, the dictionary suggests that counterpart and complement are synonyms, but in this sense of completion, complement is what gets used. If you're composing a meal and working out what goes well with what else, you are looking for good complements - things that complement each other well, using it as a verb.

If you want to maintain your use of counterpart, you can do so by using male and female as adjectives:

The male population is higher than its female counterpart.

The population of males is higher than their gender counterpart.

I wouldn't say that this is flat out wrong, but it makes the reader stop and think to understand what is being said, and it feels both akward and pretentious. I would advise against using this.

The population of males is higher than their female counterpart.

This is clearer, but "counterpart" seems to serve no useful purpose except to increase the average word length and falsely suggest greater precision.

The population of males is larger than the population of females.

This is clearer and more accurate. If it is OK to write "males", then it should be OK to write 'females". The parallelism in form emphasizes the parallelism in meaning. "Higher" not as good a way to describe the increase in population numbers as "larger".

By the way, in the example sentences in the question, either "their" should be changed to "its" treating a "population" as a single thing, or else "counterpart" should be "counterparts", treatign it as a collection of people. I would favor the 'its' form.