The other answers linked all give very valid explanations
I'd like to emphasise one more answer: interservice rivalry.
The Army-Marine rivalry scuttled one the more interesting plans for USMC involvement in the ETO: Project Danny
By mid-June 1944, the Germans were launching V-1 flying bombs at London from launch sites in Northern France. In response to the terror bombing, the Allied air forces launched Operation Crossbow. Hundreds of medium and heavy bombers pummeled V-weapon sites in Northern France, with mixed results.
To add some more muscle to the operation, Navy and Marine Corps planners concocted Project Danny. The plan called for Navy escort carriers with Marine fighter squadrons to be deployed off the coast of France. The Marines' F4U Corsairs would be armed with massive 12.75 inch "Tiny Tim" rockets and sent to smash the V-1 launchers. All in all, five squadrons of Marine fighters from Marine Air Group 51 (MAG-51) were committed to the plan. Marine pilots in the U.S. even began training in earnest for the mission.
But the plan still had hurdles to cross. Atlantic Fleet commander Vice Admiral Patrick Bellinger and Commander Thomas Moorer pitched the idea to Army chief of staff George C. Marshall. Before Moorer could even finish, Marshall cut him off and stormed out of the room.
"There will never be a Marine in Europe as long as I'm chief of staff," Marshall growled. And that was that. Project Danny was shelved in July 1944.
Why did Marshall want to exclude Marines from the ETO? Marshall had been disgruntled with how Marine Corps "Devil Dogs" had grabbed headlines during WWI, obscuring what he felt were the far larger contributions of the Army. By excluding Marines, Marshall was trying to prevent the Leathernecks stealing the spotlight again.
You may be interested in A Different War: Marines in Europe and North Africa. It includes descriptions of Marines who served in the ETO. Most served on ship detachments aboard US battleships, carriers, and cruisers in theater. There were also Marine detachments assigned to the US embassy in London, and to guard duty in Londonderry and Iceland.
More particularly to your question, a Marines gun crew was killed aboard the USS Savannah when it was struck by a German radio-controlled glide bomb while supporting the Salerno landing. There were also Marines who served in the OSS--two were killed in action in Europe and another received the Silver Star for his work with guerilla forces in Yugoslavia.