Dave,
Your gun is a FN (John Browning) Model 1900 in .32. It was extremely popular in Europe well beyond WW1. The fact that a family member carried it while in the American Ambulance in Italy makes it all the better. When WW1 broke out, America was trying to remain neutral under Wilson. However, Germany did not issue a formal protest for the volunteer American Ambulance Services. In fact, many of the now woke universities, purchased their own Model T's and converted them to ambulances and the students became the Ambulance personnel. Stamford, Yale, Harvard, Dartmouth, Princeton and others sent Ambulances and volunteers to France and Italy beginning in 1914. Many of the college students went over with the American Ambulance and joined the French Foreign Legion and some went to the Lafayette Escadrille which became the Lafayette Flying Corps when the US entered the war. The American Ambulance was an extremely small group and those in Italy are very rare. If by chance you have his uniform and paperwork, medals, photographs and any other of his material, you would have an archive that would value into the thousands. If he went over with the American 332 Infantry in 1918, he saw a lot of combat and had 30 Ambulance Sections in his infantry division. If you still have his 332 Infantry shoulder sleeve patch in gold bullion, it can fetch up to $500 for the patch alone.
As an aside, Earnest Hemingway was in the Ambulance Corps.