Wife wants to separate, but husband doesn't want to. Will divorce happen in this scenario?

It depends.

If you're looking at it from the American perspective or from the Islamic perspective.

From the Islamic perspective, the wife would have to request a khula and the husband would have to accept that khula.

The other avenue that the wife has is to request a faskh, which is a judicial dissolution of the marriage. Basically present their case to a panel of muftis and if the mufti finds that there is a basis for dissolution, then the judge can award the divorce.

There's something else called the tafweedth, which is a pre negotiated ability to divorce. But those are the ways that a wife can get a divorce under Islamic law.

But from an American legal perspective, a husband or a wife can file for divorce.

75% of cases that are filed in the legal system are filed by women. So from the legal perspective there's no impact as to who files on the outcome of the divorce and either party can file usually without any reason. Most states now have no fault divorce.