Is it permissible to look at girls in public places with the intention of searching for a wife, and not for harassment?

The Shariah permits a man to look at the woman he intends to propose to. If he is determined to marry her, he is allowed to look at her, provided there is no seclusion (khalwa). As for looking at women generally, it is not permissible, because it may lead to questioning a man's religious integrity.