Can broken water pump make the car not start

In most cases the answer would be no, it sounds like you have more issues why the car will not start.

Joemom is correct the only way the water pump can make a vehicle not start is the bearings are seized on pump causing belt to stop working but most of the time the belt will break first.

If water pump was leaking for long time, the timing belt may have been damaged and if so, yes the car won't start. If timing is off far enough the pistons will destroy the valves.

Water pump