Given a closed domestic water system (pressure regulator is installed
and assumed closed).  A 50 gallon water heater is initially filled at
a pressure of 50 psi and the water temperature is 60 degrees F.  The
heater comes on and raises the water temperature to 120 degrees F.  By
what method or formula can one calculate the resulting water pressure
(gage pressure) at the higher temperature?  I have actually measured
mine with a pressure gage-------but that's cheating.  I have searched
the net and found no answer except the 'gas laws' and these don't jive
with my readings.  I would like to take any initial
pressure/volume/temperature and figure the pressure at any higher
temperature (under 200 deg. F).  Am I missing something too simple to
be explained on all the physics sites??  Thank you!!

I am curious about your question because I was a back-up water
operator on a small water system. The operator started installing new
meters that included a check valve to prevent backflow into the system
from the customer's service. As you seem to understand, this creates a
closed system and the water has no place to expand when heated. I did
quite a bit of research on the problem and as you probably also found
it is a very serious situation. I assume that your gage readings are
lower than calculated values. Possibly the reasons calculated numbers
differ from what you actually read on your gage are:
1. You have all the volume in your entire plumbing system to expand into.
2. The water heater deforms (bulges) as the pressure increases.
3. You may have small leaks that you don't know about.
4. Your pipes also deform due to increased pressure.
I did find this interesting article:
http://www.hotwater.com/bulletin/bulletin41.pdf

There are lots of complications.  When the water heats up, so does the
container, so it expands as well.  But it has a different thermal
expansion coefficient, so it expands by a different amount.  Probably
it's made out of steel, which has a lower expansion coefficient than
water, so the pressure goes up with temperature.  But you can imagine
a water heater tank made from a material with a larger expansion
coefficent than water, and in that case, the pressure would decrease
with increasing temperature.  In the end, as redhoss pointed out, the
result of the exansion of the water is that the tank bulges.  So the
increase in pressure depends on the resilience of the tank.  If you
had a tank with inch thick walls the pressure would go up by a lot,
because the walls are so strong they hardly deform at all when the
pressure goes up.  The way the tank deforms under pressure depends on
its shape, wall thickness, and material in such complicated ways that
the way to solve the problem is to do what you've already done:
measure the pressure as a function of temperature.

I admit I don't know anything about water heaters.  But if there is
air in your water heater (I don't know if air is in water heaters)
then you will have a two phase problem which will explain the
discrepancy in your calculation.
Just a friendly comment from your Internet neighbor.