You definitely can! I don’t have a passport from my birth country so I have been using the Reentry Permit instead of the national passport to travel everywhere around the world from France, Japan, South Korea, Canada, Australia, and even a cruise to Mexico, Bahamas, Bermuda. The problem is I’m not sure if you have to apply for a visa using your Reentry Permit or not. Each country has a different law. But I have been using only my Reentry Permit and GC with no problem exiting and entering a country. I just have to apply for a visa to all the countries above except for when I arrived on a cruise.