Yes, it could.
Instead of using a weapon to make a melee weapon Attack, you can use an Unarmed strike: a punch, kick, head--butt, or similar forceful blow (none of which count as weapons). On a hit, an Unarmed strike deals bludgeoning damage equal to 1 + your modifier. You are proficient with your Unarmed strikes.
I'd certainly say that throwing someone counts as a "similar forceful blow". This wouldn't be an effective strategy due to the very low damage, and the DM (you) would need to decide how exactly would the dragon roll against two PCs (one thrown, one being thrown at). You would also need to consider the fact that the attack against the unconscious character, if it hits and the attacker is within 5 feet, is an automatic critical hit, and a critical hit equals two failed death saving throws. Thus, by doing what you described, you may kill your PC without giving his comrades a chance to heal him.
Personally as a DM I would rule that being grabbed by a dragon is not yet an attack, so no crit and no double death fail for a thrown character. If the dragon succeeds against the AC of the character he is throwing the body at, I would count both to be hit. If not, I'd rule that the character being thrown just tumbled relatively harmlessly. I'd rule that way because the player whose character is down probably is having a bad day now anyway, and finishing his character off wouldn't add "cinematic scope", just frustration.