The gravitational force on the Space Station obviously is not zero. It is gravity after all that makes the Space Station orbit the Earth.
The term "zero gravity" does not refer to the force due to gravity. It refers instead to all forces except gravity. When you go to an amusement park and take a ride that bills itself as having a zero g roll, or a zero g drop, gravity is not changing during the course of the ride. Your uneasy stomach tells you that something is changing. That something is your "apparent weight" or "scale weight".
You can't feel the force due to gravity. You can't conduct a local experiment that measures gravity. An example: An accelerometer placed at rest on the surface of the Earth registers an upward acceleration of 9.8 meters/second2. Why? The answer is that the accelerometer does not (and cannot) measure gravity. It measures everything but gravity. In this case, it measures the normal force from the ground that pushes the accelerometer upward. The accelerometer does not sense the downward force from gravitation, so even though the accelerometer is at rest with respect to the Earth it registers an upward acceleration with respect to the Earth.
When NASA and Dr. Tyson speak of the problems such as bone loss due to the zero g environment of space, they are referring to weightlessness (scale weight) rather than gravity.