What does Rick mean when he says "When it comes to women, you're a true democrat"?

I don't think it is about being "liberal" so much as having "democratic" beliefs. To say someone is democratic means they believe in equal representation. By calling Renault a "true democrat", Rick is saying that Renault is indiscriminate with his choice of women. All women are equal in his eyes, not politically but sexually. Renault doesn't have a type, he'll pursue any girl. When one jilts him, he moves on smoothly to another. It suggests his pursuit of all women is akin to a political ideology, which really means Renault doesn't have an ideology at all: he's just an opportunistic hedonist.

But I think that there's a little more going on. Rick is an American and is probably attached to democracy as an ideal. Meanwhile, decidedly undemocratic ideologies like Nazism and Fascism are warring with democratic nations to determine which philosophy will win out. By calling Renault a "true democrat" while living in the colony of an occupied "democratic" country, Rick may also be making a wry statement about what little power dissidents have.

This has absolutely nothing to do with the American political party of that name. Here, democrat (and not Democrat) is used in its regular meaning (either 1, or more likely, 2 of the definitions below):

1. an advocate of democracy.
2. a person who believes in the political or social equality of all people.
3. (initial capital letter) Politics. a member of the Democratic Party. a member of the Democratic-Republican Party.
4. Also called democrat wagon. a high, lightweight, horse-drawn wagon, usually having two seats.

The movie Casablanca is set in 1941, in the Moroccan city of Casablanca. At the time, Casablanca was "under the dual control of Nazi Germany and the Nazi-puppet government in Vichy, France, that the Germans had installed after occupying France in 1940". The Prefect of Police, therefore, was the Prefect of Police of a fascist state, the very antithesis of a democracy. He was a member of the executive branch of a fascist government.

Therefore, the line you are asking about is making a point about precisely that. Although, despite his personal political positions, Captain Renault is the arm of the Fascist government, when it comes to women, he is a true democrat: a person who believes in the political or social equality of all people. Or, more likely, one who believes in the "right" of any and all women to end up in his bed.

Rick is an American expatriate. A Democrat (at least in America) is a member of the Democratic party, which is the liberal party in US politics.

Another definition for the word liberal is "not bound by authoritarianism, orthodoxy, or traditional forms."

So, by calling him a "true Democrat" in regards to women, he's suggesting that Renault is a womanizer; someone who is "not bound by traditional forms." ie: he sleeps around a lot. This is prompted by Renault's earlier statement about paying a visit to Yvonne on the basis that "someday they [women] might be scarce." (To say the least, a flimsy excuse if ever there was one.)

I believe this account has nothing to do with sexual equality. It is clearly a slam against Renault acting in an unscrupulous way toward a vulnerable woman on the 'rebound'. Rick is a Republican evidently since he fights wars against socialists and fascists, now pointing in sarcastic tone to Renaults' weakness, i. e. Seeing another persons weakness and seeking self gratification easily in a woman unbalanced and easily dominated. Renault does this during the film, being told by his servant that a particular woman was needing help, waiting outside his office. He spruces up before the mirror. Obvious predator, using his power in office to 'trade' favors for an exit visa. It is this dictatorial power over a woman in weak position that Rick is predicating against. He fights for the little guy by nature, but in Casablanca he himself has been wounded and is off balance until his relationship is healed with Ilsa.