There are many, many methods of leverage short of war. International relations happens all the time between every country on earth; only a tiny number of those interactions involves violence. In fact, a primary job of the people who practice international relations is to get things done without risking wars - sort of like doctors whose job is to get things done without killing the patient.
> some guy who wasn't even our citizen, killed in Turkey, by a foreign government
Human rights are widely considered to be universal, both in philosophy and in law. The U.S.'s own Declaration of Independence says 'all men are created equal ...'; it doesn't say 'all Americans'. Also, it's widely believed that if you let human rights deteriorate elsewhere, you are creating more risk for yourself. If your next door neighbor doesn't have freedom, there are many reasons to think you might be next. Finally, free, democratic countries tend to be much better partners on the world stage - they tend to be peaceful, stable, and prosperous.
Finally, Khashoggi was a journalist, which implicates a specific human right, freedom of the press.