I read the Essential Punisher book a little while ago.
Assuming they were collected in the order they were written/printed, no one, including Spider-Man, seems perturbed by Frank killing people for the first few issues. And then Frank meets Captain America, who does object. After that, Frank is treated as a dangerous killer by everyone, including Spider-Man.
There was probably an editorial mandate that came down at some point, or an angry letter or something.
I sort of think of it this way: Peter doesn't object to soldiers killing at war, or police killing criminals in self-defense. He'd kill if he really thought he had no choice, though he'd feel like crap about it after and he wouldn't do it lightly. The first few times he meets the Punisher, he kind of thinks of it that way; Frank has to shoot people because he doesn't have powers or webs to get around it. (I know this makes Peter kind of an idiot but let's just use this conceit.)
But after a few meetings, it's like... dude, enough with the shooting people already. Get some help.