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.
It's not though. Lots of military ambulances, from many countries, not associated with the Red Cross, have a red cross on them.
One example, there are many more: https://www.google.com/search?q=british+military+ambulance&t...
"Under the Geneva Conventions, the three distinctive emblems of the red cross, red crescent and red crystal are intended to identify and protect medical and relief workers, military and civilian medical facilities, mobile units and hospital ships during armed conflict."
https://www.redcross.org/content/dam/redcross/atg/PDF_s/Inte... (PDF)
Edit: Perhaps I misunderstood. I thought the Red Cross was complaining about military ambulances, hospital ships, and so on, depicted in games. In addition to less realistic uses.
I suspect the red cross would not take issue with a game using the red cross to depict untargatable aid workers.