There are an awful lot of characters in Middlemarch that have unrealistic or unattainable dreams, and Mr. Lydgate is no exception. Like Dorothea, he wants to do good work in the world, and, like Mr. Casaubon, he wants to leave behind some great scholarly discovery. We can sympathize with these noble ambitions, and we can feel sorry for him when they don't work out. But that doesn't mean that Lydgate's ambition is all there is to him. Unsurprisingly for a character in a George Eliot novel, he's got a lot of layers. He's strong and determined, but easily swayed by Rosamond; he's perceptive and can sense what other people are feeling, but he doesn't easily sympathize with them. In short, he's a complicated guy.
Lydgate, like Will Ladislaw, comes from a wealthy, upper-class family, but has to make his own way in the world. Lydgate's parents died when he was young, leaving him to be raised and educated by his uncle, Sir Godwin Lydgate. Sir Godwin was nice enough to let Lydgate choose his own profession, but was kind of disappointed when Lydgate chose medicine. Unlike nowadays, when doctors are thoroughly respectable, in the 1830s, when Middlemarch takes place, becoming a doctor was considered a step down for a guy like Lydgate.
The medical profession wasn't very well regulated in the 1830s – no one had to pass boards or take standardized tests to become a doctor. Lydgate wants to change all that. Part of his ambition is to make doctors more scientific. Instead of prescribing the same crackpot "medicine" that had been prescribed for decades, Lydgate wants doctors to look at symptoms scientifically, and to take notes about what works and what doesn't. Unfortunately for him, though, other doctors in Middlemarch aren't so excited about having their practices criticized by some newcomer.
Just the fact that Lydgate is willing to take a stand for what he believes to be right suggests that he's proud (in a good way). But he's also conceited. When he thinks he's right, he considers anyone who thinks differently to be inferior. Take a look at this passage:
Lydgate's conceit was of the arrogant sort, never simpering, never impertinent, but massive in its claims and benevolently contemptuous. He would do a great deal for noodles, being sorry for them, and feeling quite sure that they could have no power over him. (2.15.9)
So Lydgate's never mean about it, but he's "contemptuous" of those he thinks are "noodles." He's condescending, but at least he's nice about it – he's never "impertinent" or rude to the stupid "noodles" of the world.
Arrogance might be a character flaw, but it's one we can live with – Lydgate's major problem is that he's not able to sympathize with people. Before his own money problems start, "he had no power of imagining the part which the want of money plays in determining the actions of men" (2.18.3). In other words, he has no sympathetic imagination. To put it like a kindergarten teacher, Lydgate can't put himself in the other guy's shoes. He's incapable of "imagining" why other people act the way they do, and that's part of why he's so "contemptuous" of people who think differently from him. Part of his lack of sympathy is that he's too scientific. He's more interested in causes and effects than he is in what people are really feeling:
[Lydgate] wanted to pierce the obscurity of those minute processes which prepare human misery and joy, those invisible thoroughfares which are the first lurking-places of anguish, mania, and crime, that delicate poise and transition which determine the growth of happy or unhappy consciousness. (2.16.56)
Lydgate is only interested in "human misery and joy" or in "anguish, mania, and crime" in an objective, scientific way. He wants to find the hidden "processes" in the human mind that "determine" whether you'll be "happy or unhappy." This might be an interesting scientific pursuit, but it makes him a crummy shoulder to cry on. If you try to tell this guy your troubles, he'll want to look at your throat or take your pulse, rather than listen sympathetically.
This is part of why he and Rosamond are so unhappy together. His character makes him unable to enter into what someone else is feeling, and her character makes her just not care what anyone else is feeling.