What gives you the impression he simply open the door and shot her?
Based on what we know as fact, I'd say it's more likely he cautiously opened the door to see what she wanted... And she ripped through the screen trying to get inside. Personally, I would've just shut the door on her arm over and over, but he's completely within his legal right to shoot her if that's what happened.
Two things. First, the screen has little damage, consistent with a shotgun blast from reasonably close but not point blank range. (A shotgun blast from farther than a foot or maybe two lacks the force wave to blow out or rip a screen; a shotgun blast from less than a foot is going to blow a round hole in the screen from concentrated muzzle blast regardless of whether the screen is pulled loose.) Therefore we know the homeowner was likely backed away from the screen. Probably he was close enough that the muzzle blast ripped the screen but not so close that it would remove a round plug, given the police statement that he fired through the screen, although it's also possible that she ripped the screen loose. Too close and a round section of screen is removed; too far and the only damage is from individual pellets. If the rip is caused by the muzzle blast, then the distance will be known to a foot or two; otherwise the density of the pattern (where shot strikes the screen) does much the same.
Second, we know from police statements that the girl was not shot from "very close range" and that she was shot on his porch, not in his house, and that the shot was not at an extreme upward trajectory. Since the homeowner was not immediately charged with possessing an illegal firearm, we can assume that his shotgun has a minimum barrel length of 16", meaning even a stockless home defense pistol grip model needs to be at least arms's length from its target. Adding barrel length and the specified shot distance range, had he opened the door and she began trying to enter, either she'd have been inside when he fired or the shot would have been point blank and probably contact. He simply could not physically prevent her from entering without the shot being point blank, if not contact, with a legal length shotgun. Assuming normal home defense barrel length of 17" - 20", swinging open the door and leveling the shotgun at her head would be point blank were she standing at the screen, so I'm assuming that either she was at the screen but was not quite so fucked up as to not back up when a shotgun was aimed at her face, or possibly that she had moved away from the door but began moving toward the door once he opened it. Another possibility is that he had backed away from the door, so that the rip was either pre-existing or possibly her damage. In that case she could have been standing at the screen, possibly even trying to get through it. But in that case he probably could have claimed home defense even with her, so I suspect she was backed away from the screen and his shotgun was within a foot, two at most, of it when he fired.
Your opinion or do you have link to the homeowners statement to police that backs this up?
Totally my opinion. I just don't believe that a man who answers the door with a shotgun (which I am not against on principle) would ever sleep with the door open and only a fragile and easily defeated storm door screen keeping the bad guys out of his house. A burglar or home invader could bust through a screen in just a few seconds, or cut stealthily through it in a minute or two without awakening anyone. Those two behaviors are totally at odds, as sleeping with your door or ground floor windows open implies a confidence that nothing bad is going to happen whereas answering the door armed implies the opposite.