Wearing a Red Cross while shooting at someone is a war crime.
Which is why combat medics, who are soldiers and who are expected to shoot at people if required, do not wear the red cross. Other medical personnel whose role does not include combat, like ambulance drivers and paramedics, often will.
The red cross indicates that someone is a humanitarian worker and a non-combatant. That's why shooting at someone while wearing one is against the Geneva conventions. It is entirely within the Geneva conventions for a medic (someone with medical training) who does not display the red cross to carry a gun and shoot at people.
Again, the red cross is about displaying whether or not someone is a non-combatant, not whether they are a medic. This wasn't true to the same extent in world war 2, but it has been true since.