Yes, but it's unlikely
It's entirely within the rules to continue attacking a character while he's unconscious, and a GM/monster might decide to do so for a number of reasons. However, there are usually more reasons he won't, both in-game as a monster decision and as a meta-game GM decision.
First, there's a rule
Though more what you'd call a "guideline" than an actual rule.
Don’t hit people when they’re down. When a character falls unconscious, monsters turn their attention to enemies who are still up and fighting. Monsters don’t usually intentionally deal damage to fallen foes. [DMG 40]
So GMs are clearly discouraged from kicking people while they're down... but why?
It's a poor tactical choice, or at least makes sense in context
When a PC is already down, the monster is going to be much more concerned about the people who are still able to deal him damage and debuff him.
If the monster has some RP reason to particularly hate the downed PC (nemesis type villain, maybe?) then it could be better justified as a tactical choice—but it's still not a good one.
It's a vicious—and frankly boring—GM choice
You've spent time and effort on your character and unless you're in a group that has decided ahead of time to be excessively lethal, a GM using coup de grace should generally have a good reason it won't make you hopping mad as a player to get taken out so undramatically. It'd be much more interesting for the PC to get dragged off to the monster's lair / laboratory / tropical resort, setting up scenes of dramatic rescue or escape (this is actually suggested in the paragraph following the above DMG quote).
But sometimes it's a good tactical choice, or at least makes sense in context
For example, the Fell Taint monsters heal to full if they successfully kill a creature with a coup de grace. They trade this for dropping their insanely strong defenses (insubstantiality) and being unable to move for some time, so it's tactically interesting.
And not all monsters make good tactical decisions, of course.
In these cases the GM should consider the "vicious GM choice" aspect when deciding if the zombie is going to stop to chow on some brains in the middle of combat, or the evil mastermind takes a moment to skewer an unconscious foe.
And sometimes it's a good and interesting GM choice
PC death can be good for story and character growth, provided the GM and the players trust each other to make decisions the group finds palatable.
Addendum: "Kill him before he heals again" isn't a no-brainer in 4e
In 4e, NPCs and monsters are mechanically distinct from PCs. One of the results is that no NPC has more than one healing surge per tier, and very few have any more healing powers than Second Wind.
This means that unless a monster has prior knowledge of the PCs it has absolutely no in-game reason to expect they'll be able to heal dire wounds repeatedly, or that a particular character will have the ability to grant multiple massive heals to others. Within the 4e ruleset, the instant mid-combat healing of grave injuries is not common: it is the purview of player characters alone. Not even gods have access to the breadth and depth of healing-per-day that level 1 PCs do. This is part of what makes them heroes: their access to unmatched ability and potential.
Of course, a villain that has fought the PCs before, or who has observed or researched them, can reasonably know this. A monster that sees the PCs go up and down like yo-yos may get tired of it and adjust his tactics toward the end of the battle.
Coup de grace?
4e has the coup de grace action, which allows any successful standard-action attack power that targets a helpless creature to be an automatic critical. Regardless of whether the damage dealt should kill the target, the target dies if the damage is at least its bloodied value.
There is no good reason for a monster attacking a helpless creature with lethal intent not to use this action, aside from being blindingly stupid or some similar handicap. However, a GM might choose to 'forget' this rule if he wants to attack a helpless player to indicate the seriousness of the monster's threat without practically guaranteeing a PC kill.