How can I calm panicked units?

Panic is a one-shot deal. When a unit panics, they react erratically, and lose their next turn.

This means that, in a best case scenario, you will regain control of the panicked soldier on the next turn.

However, XCOM being what it is, your soldiers may panic again (the lowered will from having already panicked not helping), with the same unpredictable results.

Late game, the psionic ability "Psychic Inspiration" will remove panic, and re-allow them to take their actions. If cast preemptively, it can often prevent panic from happening in the first place.

In your specific case, if you have any soldiers left standing I would use any and all means necessarily to eliminate threats and hope for the best. If you can win the mission I think you'll be ok (I had a squad mate bleeding out when I won the mission and he survived. Wounded, granted).

I've had panicked squaddies calm down. I can't say whether this was due to morale settling "naturally" or if removing immediate threats helped with this. I don't know if later-game technology will help with this, but there is nothing I know of in the early game.

@donnniegoat provided a comment highlighting this PC Gamer article about a "Psychic Inspiration" ability that will calm panicked squaddies by boosting willpower and removing panic and mindfray, but unfortunately it won't be available until well into the game.