What do you call the measurement of the gravitational force acting on an object?

Basically it is the object's "weight".

The gravitational force on an object is its Mass X Gravitational Constant.

The gravitational constant is the acceleration of a free falling body towards another body, and on Earth is equal to 9.81 meters/sec2 or 32.2 feet/sec2.

Thus while the MASS of an object is a constant physical property, the WEIGHT of an object depends on the local gravity field pulling on that MASS.