I've fallen and I can't get up!

You can tell Apple this is a feature you'd like to see in a future release, but currently, there is no way I'm aware of this can be accomplished --> Feedback - iPhone - Apple

If there were......I cannot think of a way for the phone to know the difference between sitting on a nightstand next to a bed while you are sleeping or resting on a table while you are napping, from sitting on the floor after a fall.

As Bob said, this would be a bad idea since the iPhone could place emergency calls potentially multiple times a day when nothing is wrong, the phone is just sitting on a table. You can, however, get an Apple Watch which has fall detection and can call emergency services automatically. Or there are dedicated devices on the market that can do something similar.

Regards.

Why not set a repeating alarm to remind you to send your son a text or call him or email him at a specified time or two every day. The alternative would be a fall detection and notification device. There are services starting at about $40 a month.

You're welcome, but I think your comment wasn't fair. Everyone who responded to you understood what you were asking. It's just not a feature which is on iPhone at this time. Jeff Donald offered a good suggestion, which is to look for another type of tracker, which may have this functionality built into it, since iPhone doesn't.

Then, the answer is "no".

Are you trying to use it for fall detection or just a wellness check after a specified period of time?