Your genuine US Navy uniform (not a Fashion/Imitation) button's exact backmark, including the two stars and two rings of dots, is shown in a photo in the book "American Military Button Makers And Dealers; Their Backmarks & Dates." The authors say your specific backmark was in use only from (circa) 1860 to 1870.
You didn't ask, but somebody will want to know:
Being an 1860-70 backmark, it COULD have been made during the civil war, but equally possible to be from 1866-70. In such cases we have to rely on the dig-location as an indicator. Being found in Spokane WA gives us no help. BTW, even when an 1860-70 button is found in an area of civil war military activity, it MAY have been made in 1867 and came off the uniform of a yankee Occupation soldier or sailor during Reconstruction. That is why we Virginia diggers find "absolutely-definitely postwar" yankee buttons around Richmond.