Is there a manual override on the fixture that would let you know if the thing is working at all? It doesn't seem to me that you would wire it incorrectly, there's only two wires plus a ground. I suppose you could try reversing the hot and neutral. I'm assuming you've checked to ensure you're getting power to the fixture.

The next step might be to try one of the fixtures that you know is working.

the light does turn on when motion is detected so it is working - the problem is that it will not stay on like the other 2 lights do.

Do the other 2 lights shine on the third's sensor? Because maybe the third light doesn't think it's dark out, so it turns off.

I wish that was the case - only one of the other 2 lights could shine on the one that does not work and I took the bulb out of it. The nonfunctioning light still does not turn on.

Many motion lights have a switch or dial to select the "on" setting. The one light may not have "dusk-dawn" selected.
If your replacement also won't function like the others, it's unlikely that the problem lies in the fixture. Probably an issue with its settings or its location.

this light does not offer this option - it only turns on at dusk. There are different options as to dim versus bright, but no on-off option.

The motion detectors that I have use some combination of flashing the switch to change modes of operation. Do you have the instructions still? If not, how about the manufacturer and model number?

I can't find anything in the instructions.

The unit is:

New Haven Smart Lighting 9 inch outdoor wall light dark bronze lights (product sku SR53131OR)

Here's the instructions:
Manual

Looks like there's a switch on the bottom that controls the timing of 3 or 6 hours after dark or dusk til dawn (the '0'). You are probably in test mode (I think that means the motion feature works whether it's light or dark out, but the light sensor does nothing).