Can a broken circulation impeller cause a dishwasher not to drain?
The impeller is the main part of the pump. The pump makes the water spray and it also pumps the water out after the wash or rinse cycle has finished. Usually the drain solenoid opens and since the pump is running all the water goes out the drain instead of through the sprayers.

It doesn't drain by gravity since it pumps up to the sink tailpiece or other entry to the plumbing drain system.

Once you repair the pump all should work as normal but you will have to drain it by hand before repairing it or dump the water all over under it when you open it back up.

Note: some dishwashers have two impellers: one to spray the water during the wash cycle, and one to drain the water. Replacing the spray impeller will not fix drainage problems. If your drain impeller is fine, either your drain line is blocked, or your motor has problems.

Good luck with your project!