Does "going to college" exclusively mean getting an undergraduate degree in the US? "Graduating from Duke" and "completing a medical degree" both sound inherently going-to-college to me.
Here in Australia we'd sub in "uni" for "college", but it just sounds weird to "complete a degree" without "going to [tertiary education]".
Without thinking about it too much, I think I'd say "going to college" for undergrad (even at a university), "going to grad school" for a graduate degree, and "going to med school" for a medical degree. (For reference, I'm from Michigan.)
In the US a significant proportion of every undergraduate degree is "general education", ie what in Australia would be an Arts degree. AFAIK, specialized degrees like law and medicine are exclusively post-graduate degrees.
Here in Australia we'd sub in "uni" for "college", but it just sounds weird to "complete a degree" without "going to [tertiary education]".