The medic appears in a number of different squads. For this guide I am only showing the rebel faction but the same principle applies for all factions.
The below picture shows the Rebel Support Team consisting of 3 soldiers and the Rebel Veteran squad consisting of 5 soldiers. The squad icons highlighted with the yellow box show a medic symbol or + symbol. This tells the player that there is a medic in this squad.

You can quickly identify who the medic is by selecting the entire squad and looking at the squad silhouettes located down the bottom under the quick action bar. In the above pic you will notice the silhouette i highlighted in red at the bottom of screen has a medic symbol telling me that he is the medic soldier for that squad.
In higher tier squads, each member in the squad has a medic capability.

All soldiers will automatically start healing themselves if they have personal medic kits. They will heal minor wounds until they have either expired all their personal medic kits or their health bar is back to full.
Soldiers that have lost too much health will lay on the ground and have a white diamond with a red blood drop on it telling you that the soldier has been mortally wounded and needs a medic. The pic below shows such a soldier. This soldier will eventually die if he is not given assistance by a medic

To start healing him, select your medic and right click on the wounded soldier, the medic will run to the soldier and commence the healing animation as seen below. If you click on the pic to make it larger you will see a small red health bar. This bar indicates the progress of the healing action and once it is full, both the medic and soldier will lay on the ground until given further orders. This is where you should quickly remove your medic from danger and get the wounded soldier back into action.

The medic has enough supplies to heal more than one soldier, how many exactly i am yet to determine but i think it is based on the amount of health kits in his inventory. The medic generally carries twice as many medical kits than a normal soldier.

That concludes the medic guide. I hope it helps you.