Would ruling another nation be compatible with basic American ideals of democracy and self-government?

No, the act of ruling another nation would be Imperialism, which goes against the principles of democracy and self government. Temporarily occupying a country after a war, such as the United States did after defeating Japan, came close to ruling another country, however, this was only a temporary arrangement until they had set up a new constitution, and Hirohito had denounced his divinity.