Throughout the entire history of the U.S., Russia and the U.S. seem to have had a decent enough diplomatic relationship until the end of WWII and the beginning of the Cold War. From what I have read Russia even had a part helping the U.S. in the first Gulf war with Iraq. Have the two countries ever directly through military action caused each other casualties?
So one event we can point to is the Allied Intervention in the Russian Civil War.
In the summer of 1918, Wilson was convinced to commit US troops to initially try to keep Russian troops in the field against Germany and reopen the Eastern Front, in order to keep any more German troops from being used in the West. And to keep up political pressure on Germany. Additional considerations were to try to solve the problem of the Czech Legion stuck in Siberia and who had seized parts of the Trans Siberian Railroad. And at least on the part of the British, driven by Churchill, to kill Bolshevism in its cradle as Civil War between the Reds and Whites broke out.
In the Summer and Fall of 1918 some 15,000 total US troops were landed as part of Expeditions in Siberia and in North Russia. While other allied forces like Britain took a much more active role in positioning and fighting, US forces mostly guarded supply posts, communications routes, and tried to enforce general order.
After the Armistice, and escalation of the Civil War, breakdown in the White forces unity, and plummets in Morale US forces were then withdrawn over the course of 1919.
In total casualties were a bit over 400, of which a good 40% was from diseases like the ubiquitous Spanish Flu, but also about 100-150 can be attributed to battle, almost all of which was with Red forces.