I don't think that this rule is relevant any more. It comes from the days when it was seen as good manners to be excessively courteous to women in public: men opening doors to to let women pass through first, men not sitting down until women are seated, and so on. This is all fairly outdated and it makes most people feel uncomfortable. Nowadays, the only rule for introductions is if one person is obviously higher status than the other. For example, a junior person in a business context would be introduced to a senior person - not the other way round. Otherwise, it really doesn't matter.

In the US, either way works.