Can a depressing character be likeable?

My current protagonist, Nicholas Blade is a milestone for me in terms of storytelling. He's the first protagonist character who I've written about that seems to have no value in his or many others lives. I've had many other characters like him but none have been in the main character position before.

I drew influence from one of my favorite Final Fantasy characters ever Squall Lionheart, because he too was pessimistic and bleak. While he wasn't on the level of Nick he still hated social interaction and people from the start, but as the story goes on you witness him break out of his shell and learn to accept others.

That's sort of my goal with Nicholas, throughout the series I want to make him value himself and others as his life gets jerked around in several crazy and even supernatural ways. I want to paint the painful voyage of self growth and realization that would fascinate the reader into wanting to stick with him.

But my problem comes from a relevant point in a review from the webseries Zero Punctuation where Yahtzee criticizes the main character from the game Darksiders because "he doesn't give a toss about anything he does" and added "so why should I?" Because from the earlier stages of the plot line Nick can't be considered likeable he's often portrayed as actually quite messed up

Example:

 I gave a quick sigh and rolled my eyes. Chris is always insisting I patch things back up with Veronica, my ex-girlfriend and mother of my small daughter, Anna. I’m not sure why he keeps prying at it, he doesn’t seem to grasp why we separated, come to think of it I don’t even know anymore. She was perfectly fine with being with a dangerous vigilante for two years, but it was in the third year when we were expecting Anna she started changing heart. She became more and more involved with the church and their strange ways of fasting, prayer, what have you. She tried to convert me over to all of it, but I knew better than surrendering myself to that brainwashing, that's why I left the church a long time ago. I guess that’s where it all started to fall apart for us, I remember she gave me the ultimatum not soon after Anna was born: either I give up this life or I give her up.

The choice was much easier than I thought. Its much easier to not grit my teeth at every prayer she did for my soul, its much easier to not have to hear her cry over whether I’ll die the next day because I am what I am, its much easier to not look at every phony priest, do-gooder, and “tortured soul” she began to hang out with, its much easier to just walk away isn’t it? I made my choice and I won’t look back even for a second. In my opinion it’s much better to live and die alone in this world anyway. That way there’s nobody who’ll miss you, nobody to cry at your wake and speak about you in past tense, nobody to pay for your shortcomings in life, you don’t leave anyone else with your burden, you just disappear. And that's perfectly fine with me, its weird that no one else sees it that way.

I intended on Nick being a cold bastard from minute one and progress him into a better person as he forced to interact with others in a meaningful way, but my question is could his callous actions possibly turn the reader away? Is there a point where the reader gets fed up with a character not progressing enough?