In earth grounding a scan/ham radio setup, I'm supposed to connect my antenna ground to my house panel ground. Can I connect my antenna ground to my ac outlet ground (where my outdoor cables enter the house) or do I have to run a direct line halfway around the house to connect to the main ac panel ground?? Thank you!

You should follow the antenna installation guidelines as closely as possible...
I won't comment on the operational side of this connection but as a power engineer, connecting a lightning rod (your antenna looks like one) to a receptacle would route a lighting strike through your house. Making the connection at the grounding electrode system as close to the electrode (likely the copper incoming water service) as possible would be my recommendation.

You're talking about grounding the metal structure of the antenna mast, correct? This ground connection is a safety feature that insures (a) there is no significant difference in voltage between the antenna metalwork and ground or ground of your electrical system system and (b) that in case of lightning strike the energy of the lightning bolt will be carried safely to ground without going through wiring in your house. Attaching the antenna mast to a ground wire on an AC outlet will alleviate (a) but would be terrible for (b). You must ground the mast to an appropriate ground rod outside the house to ensure that in case of lightning strike you don't burn your house down, blow out all the electronic devices in your house, or (heaven forbid) electrocute anyone in the house.

One option: drive a ground rod near the base of your antenna and tie the antenna metal structure to it, BUT ... that second ground rod MUST be tied to the main ground rod near your electrical panel with a 6 AWG or larger conductor. So one way or another you're going to have to connect to the existing ground rod.