I have furnace/AC made by Comfortmaker, model RPJ II with a White Rogers Digital Comfort Set II thermostat.

Furnace comes on, but not fan when on auto. When I turn fan on manually using thermostat all is well.

Is this a thermostat problem or fan control on furnace and can I fix?

Most likely a fan control problem. The t-stat should not control the fan in the heat position.

I agree with joshdta. The furnace turns the blower on in heating mode. The thermostat in cooling mode. Sound like you have a control board that's gone bad.(time delay on the blower heat relay is bad.)