Rorschach had a line in the book that summed up both him and every other anti-hero in existence. Incidentally (or not), it summed up the same point that was the driving force behind Dark Knight Returns (which is, essentially, Frank Miller's version of Watchmen anyway).
"Understood man's capacity for horrors and never quit. Saw the world's black underbelly and never surrendered. Once a man has seen, he can never turn his back on it. Never pretend it doesn't exist. No matter who orders him to look the other way. We do not do this thing because it is permitted. We do it because we have to. We do it because we are compelled."
Compulsion to put yourself in the midst of danger isn't necessarily conducive to seeking out your own death. It just means your drawn to a hazardous line of work. Admittedly, Rorschach left little room for self-maintenance, but that probably had more to do with his crime-fighting objectives than it did with assuring his own eventual death.
But there is a big difference in mental health between Batman and Rorschach, probably best underscored by Miller's re-writing of Batman in All-Star Batman and Robin.
Regular Batman is borderline depressed, but thrives. He works on his fitness and health. He lets the machine of his parent's wealth go ticking along. He has something of a life outside of avenging his parents, albeit one which at times is portrayed as a persona to cover the traumatised real personality. He has good friends and the occasional romance. In Grant Morrison's Batman and Son tpb, Bruce Wayne talks to a romantic interest about his parents' death, and says, "I got over it". But the artist (Andy Kubert) has drawn him with haunted, shadowed eyes. He clearly has not go over it, is traumatised and obsessed, but he is clearly functional. (I sometimes think that the character should marry and have his pregnant wife shot dead, so as to provide a new and fresh cassus belli, but that is another discussion.)
All-Star Batman is on the other hand psychotic. He lurches between aggression and despair. His voice sounds menacing one minute, on the verge of tears the next. He does not indulge in hygiene. He put himself in a cave as a teen and ate bats as a form of mental flagellation. In order to teach Dick Grayson about pain and loss, he tries to do the same thing to Grayson (but he is rescued from this torturous imprisonment by a very troubled Alfred, who serves as Bruce Wayne's conscience more than anything else).
Rorschach is more down this end of the psychological spectrum. He lives in a rundown apartment in a bad part of town. He doesn't seem to work much. He rarely eats, and is happy to indulge on the contents of Nite-Owl's fridge - raw eggs, cold beans from a can. He smells and Nite-Owl can barely shake his hand because his gloves are so manky - no hygiene at all. He regards his mask as his face and screams when his face is taken by the police. He can barely recognise Nite-Owl as a friend - his personality makes it very hard to recognise what a friend is.
Rorschach is a piece of wire twisted too many ways, by his awful childhood, by being confronted by the horror of the murder-kidnapping of the small girl.
Rorschach is a compelling character because of his Nietzchean will to power (Moore alludes to this with the reference to Nietzsche's famous abyss quotation). Rorschach is single-mindedly determined and intelligent (he can break into a heavily guarded base to talk to Dr Manhattan), and through hard detective work discovers the identity of the murderer of the Comedian - the smartest man in the world dabbling in some indulgent payback. Most importantly, he doesn't back down on anything: while his peers compromise, he simply cannot. He would rather die than compromise.
Manhattan killed Rorschach because there was a fair chance he (the most resourceful of all the characters) would make it home and tell everyone what happened. Did Rorschach put himself in the position of being killed? Almost certainly, knowing who he was up against. Did he have a choice in that? Absolutely not. Ducking the confrontation would not have been in character. As Ahab said in Moby Dick, "I'd strike the sun if it offended me." As Ahab fears and compromises on nothing, so too does Rorschach. And if he sheds tears as he is disintegrated, it is only through frustration at knowing he has reached the endgame and has no options left.
Rorschach is one of the best characters in comic books because he is brilliantly painted as a quite horrible, brutal, vengeful person who the reader is compelled to admire for his titanium principles. But lets not idealise him as some sort of survivalist icon, or think he should have had some other destiny in the plot.