Got back from vacation and refrigerator was out. When testing the outlets it reads "No Ground". Replaced the GFI outlet and still reading "No Ground". The breaker never tripped. All wires on the outlets to that circuit are secure. What could be the problem?

First, You don't want your refrigerator on GFI. I would turn Power off, install regular receptacle. May see ground off there, and check ground in panel.
Follow wire from frig breaker to exit point of panel, determine that ground, and verify wire is Grounded

Some early homes did not have a ground wire, if you have a metal electrical box with metal conduit attached to it you could hook the ground here at the receptacle. If no conduit it would not do any good to hook to it and you would have to run a wire from the receptacle to the breaker panel.

The ground may be missing because the wiring system does not have one, then this is why the GFI is there. The ground is not needed for a GFI to work properly.

If the system does not have a ground wire, then leave the GFI. If there is a ground wire, lose the GFI and install a standard outlet for the frig.

As far as the power, there should be 120 volts across the black and white wires at the outlet. If not, the problem many be a defective breaker, or a loose connection anywhere in the circuit from the panel to the outlet, in any box the wiring passes through. Opening each box, including the panel to check each connection and using a voltmeter to check for voltage is the only way to restore power to the frig outlet.