Note that this isn't normal copyright, the red cross is protected under the Geneva Conventions
https://en.wikipedia.org/wiki/Emblems_of_the_International_R...
A green cross would confuse people.
But tv, movies, games, etc, just showing an accurate representation of real life, within the media itself seems...normal.
I think if they have to, the least confusing thing would be a red square with a white cross inside it. That's what a lot of first aid kits seem to do. Though perhaps the Swiss wouldn't be thrilled. Maybe a white cross in a red circle?
In real life, the symbol is pretty strictly limited to a specific international organization well-known for their neutrality in armed conflict. Portraying it as just meaning 'healing' takes away from the power of that neutrality.