Although it's not perfect, I like the story and for an short film it could work fine. I think you can be a lot more subtle and a lot more visual in your story though. What do I mean with that? Just some examples: you have 'mom' in there three times. Is that necessary? Do we even have to know that it's his mom? It can just as well be a girlfriend, wife or sister or a friend. It just has to be someone who cares about him. I understand that you want to make clear the relationship they have, but don't give it.

Then you explicitly state the skin cancer thing, which is also very straightforwardly told. Try to do exposition through visuals. He is rubbing his arm where the melanoma is because it hurts and he is stressed. Then at some point he suddenly realizes what he is doing and remembers he shouldn't. You don't need to state what it is he has, he has something and he is waiting for results. That is enough for the audience to infer why he is stressed.

The two exchanges with the woman and the old man, I see what you are trying to do. He is a bit of a dick, but it's because he is nervous. But right now its a bit too laid on. The exchange with the elderly man, should be without words. Aaron walks past the man when he's on the phone and ignores him and says no with a hand gesture. A shot of the elderly mans face and a shot of Aaron's determined face would show it way better imo.

On a more general note: you can also make the character development more clear and you should define the character for yourself in the script. You could for instance make this guy a suit. Make him a businessman that works near central park and gets a cup of coffee while he walks through the park to clear his head. He is always on his phone etc etc. Then when he faces his mortality, he makes a transformation from the materialistic business man, to a more spiritually minded person. See that is a character arc.

Your film then becomes: establish materialistic workaholic douchebag, establish his fear of dying, make him face his fear of dying by getting the call and then you see him deal with it by accepting that there is so much beauty in the world.

Anyway, these are just some thoughts, hope they can be of some help.