If someone asks to stealthily open a door but rolls poorly, can they just choose not to do it, and let someone else open the door?
The roll represents an attempt to do something; the result of that roll, the result of that attempt. (See PHB p.6, "How to Play.")
As such, you can't choose not to attempt to open the door: you've already made the attempt.
I don't know how to say this without sounding snarky, but I do mean it in a helpful manner: there are plenty of roleplaying games where dice don't decide/influence the outcomes of characters' actions. If you don't want to live with dice deciding whether you're sneaky, maybe a different game is the right way to go?
[Reproduced here because, really, it should have been part of the answer and not a comment all along.]
No, and the DM can help weave this into the narrative by saying something like this:
A: "I stealthily open the door."
DM: "Okay, you stealthily open the door. There's an orc in there. Roll Stealth to see if you're stealthy enough that he doesn't notice you."
This is a good policy in general: you don't roll to "enter stealth mode", you roll when you're in stealth mode and you need to know if you evade detection.
Action declaration -> Action taking -> Action resolution
All games (not just role playing games) have at a fundamental level the concept of the player declaring/deciding on an action then taking an action and then that action being resolved to change the state of the game.
There comes a point where there is a transition from declaring/deciding the action to when that action becomes irrevocable and resolution inevitably follows. In chess, it’s when you move a piece and take your hand off it, in soccer it’s when you strike the ball or are dispossessed, in most role-playing games it’s when you roll the dice.