Can registering a business in the USA get me a residence permit, or do I have to apply for an H1B visa instead?
The USA has some investor visa options so you don't have to go the H1B route. You will need substantial amounts of money to invest. (And other conditions apply, including you must employ a number of local staff. See Patricia's comment below)
There is also an E2 visa but it is more limited in what you can use it for and who can qualify.