A little broader than that. The entire Capsicum family -- from bell peppers to chili peppers -- was introduced to the Old World during the Columbian exchange.

Black pepper is an entirely separate species, and is indeed native to southeast Asia.

The Indian pepper (black and white) being the original, and the totally unrelated American plant apparently having got the same name in English just because it, too, can be very spicy. It can get rather confusing to people who aren't native English speakers. My guess is, the GP missed the plural -s in your comment, and lacking any other attribute like "bell" or "chili", it's no wonder IMO they thought original Indian black/white pepper was what you were talking about.

(It's enough to make you wish English could come up with some other name for the damn things. Come now, she boasts the largest vocabulary of any language -- is there any sensible reason she can't come up with some other name for the paprika fruit than "pepper"?!?)