Can I connect my Apple Watch to my iPhone if I have a different provider?

Do you have a GPS +Cellular Watch? Is that what you are referring to? If you have a GPS + Cellular Watch and wish to activae the cellular on the Watch, it has to be with the same cellular carrier as the iPhone, and the carrier has to support the Apple Watch. Otherwise, if you have a GPS only Watch, you can pair it with your iPhone and use it whenever you have the iPhone close by. See this, Use Apple Watch without its paired iPhone - Apple Support

If it's a cellular watch the phone and the watch have to be with the same network provider.

yes