tl;dr: It doesn't look like they actually changed the outcome, but there's no hard numbers on gender differentials because men and women had their votes mixed, so we're not sure whether there really was a gender divide. According to most studies and research on the topic though, it seems like there wasn't a gender divide, and the woman turnout wasn't substantial enough to affect anything even if there was said divide.

So I'm just going to break this down a tad, and see what I can help you see with trends on the basis of political outcomes this election versus the 2 prior. I've heard that the mods typically don't endorse information coming from non-experts, and I'm only a political science undergrad student, but hopefully they'll see how much research I put into this and it'll be fine.

So first, let's get out the way the general voter trends in the 2 previous elections. In the 1920 presidential election, Harding won with 60% of the popular vote, and roughly 76% of the electoral votes (there were only 531 electors, as opposed to 538 now, because Alaska and Hawaii had not become states). This type of win is generally considered a "landslide", because of the disparity in electoral votes (Cox only won 24% of the electors, after all, and only 34% of the popular vote).

Comparing this with the 1916 election, we can see that the margins were much closer. 52%-48% in the electoral vote is not a wide margin, though it is less close than Bush v. Gore in 2000. Note, as an interesting tidbit, that this was one of the last times I could find in the past ~100 years that the electoral votes so closely matched the popular vote. Even the Kennedy v. Nixon election of 1960, which had a 0.1% margin on the popular vote, had 90 electoral votes more for Kennedy than Nixon. Just an interesting tidbit.

As for the 1912 election, that was nowhere near as close as 1916. With 82% of the electoral vote and 42% of the popular vote (next closest had 27% of the popular vote, and ~17% of the electoral vote), we can see that the election wasn't as close as 1916. So, what changed between all three?

Well, I'm going to take a look at the demographic breakdown, to show what groups changed. I won't pretend to know why they shifted, as that takes a lot more research, but we'll see who would've won without which group by it, hopefully.

In 1912, the win went handily to the Democratic candidate, Woodrow Wilson. Between 1912 and 1916, World War I broke out, and was ongoing. This didn't contribute to lower vote counts; there were still 3 million more ballots cast in 1916 than in 1912 (from 15 million to roughly 18 million). By 1920, the amount of voters jumped from 18 million to nearly 25 million. How many of those were women?

Well, keep in mind that we have a few things to look at. Remembering that states can (in their state constitutions) oftentimes guarantee additional individual rights that the federal constitution does not guarantee (such as gay marriage rights we are seeing now being granted in state constitutions, even though the federal constitution doesn't), there were states that allowed women to vote in 1916. The Guardian estimated the number of women voters (at the time), however, to amount to a total of 2 million. However, as the NY Times noted back then, some states did not separate between women and men voters, so we can't be sure which way they leaned. That would've been enough to tip the scales in 1916, though they probably did vote on both sides. To what degree, I can't say, but maybe it was more for Hughes, maybe more for Wilson; either way, they could've probably broken a few states to either side, if they switched to the opposite. 15 states, after all, were won within a margin of 5%, to illustrate just how close the race was everywhere. Even 1 million women voters would've comprised 5% of the total voters, and that would've been a pretty decisive vote if they all went one way or another, no matter how they were spread out over the states (barring some unusual circumstance of them all being Republican in states like South Carolina (97% voted Wilson) and Mississippi (92% Wilson). Which couldn't have happened, as they were a group larger than any individual state, and even evenly spread over all the states, they'd have made a pretty big impact in the larger states.

Regardless, women had only been able to vote in around 9 states by 1912, all Western states. This, from all the research I've seen, didn't change until 1917 when numerous other states included women in voting for presidential elections (besides Illinois, which let women vote in presidential elections in 1913, but not other elections). So in 1916, women could only vote in 10 of the 48 states voting for president, and in 1912, only 9 states of 48 (at least as far as I can tell).

The explosive increase in voting ability that followed women's suffrage is clear. 7 million votes don't just pop out of nowhere, right?

In 1920, a NY Times article estimated that only 1 in 3 women voted in the election. This, however, paled in comparison to men voting, they estimated at 2 out of 3. This was a slight drop, however, from 1912, where they estimate that ~71% voted (a 5% drop in men's turnout). They did estimate between 28-30 million voters, when it was only 26.8 million though, so what were the more official results showing?

Well, I took a look through some political science articles on JStor, and found that apparently, women didn't form any sort of monolithic voting bloc in 1920.^^1 That means they largely didn't line up with either party.

According to yet another source I found, women voting rates largely didn't match men's until 1980, so they were definitely not voting in as large of numbers.^^2

If, as the NY Times estimated, men voted at a rate 30-45% higher than women in 1920, then only very close elections could've been affected by the women's vote.^^3

All this included, I'd say it's probably a myth that he won because of the women's vote. The difference was far too significant to be laid at the hands of women, and while the vote might've been closer than otherwise, it likely was not changed in outcome.

As for your second question, I know it's technically been answered by my previous statements, but they didn't separate male and female votes in 1920. That means we don't have hard voting statistics. However, all appearances seem to suggest that women really didn't turn out enough to alter the vote, and that they didn't unite behind any candidate.

^^1 Female Ballots: The Impact of the Nineteenth Amendment Sara Alpern and Dale Baum The Journal of Interdisciplinary History , Vol. 16, No. 1 (Summer, 1985) , pp. 43-67

^^2 Women and Political Participation M. Margaret Conway PS: Political Science and Politics , Vol. 34, No. 2 (Jun., 2001) , pp. 231-233

^^3 The Gender Gap and Women's Political Influence Carol Mueller Annals of the American Academy of Political and Social Science , Vol. 515, American Feminism: New Issues for a Mature Movement (May, 1991) , pp. 23-37