Short answer no.

French and Belgian civilians could and did volunteer to work for the Red Cross who helped run some medical services, but wounded soldiers would be cared for usually within their own military's medical services.

During the early war before it bogged down if a wounded soldier was missed after a battle it wasn't unusual for a soldier to get picked up by civilians for a short time, but in the end they were returned to their own armies.

Soldiers would be cared for in hospitals either in France or England depending on the severity of their wounds. Once out of hospital they were sent to a camp run by their battalion for recovering soldiers. There they'd have their strength built back up and get more training as well as some leave and then eventually be sent back to their unit.