We entered a small parking lot from a main road ("T" shaped lot with top of T being the parking area and the short stem of the T being the entrance for 2 way traffic). We stopped at the parking lot entrance due to another vehicle approaching from about 100 feet away on our left. There were no stop signs or pavement markings in any direction. The car approaching on our left was also on the side of the parking lot that we wanted to turn into.

As we waited for that car/driver to either stop or get into their lane and continue past us, the other driver was looking over their left shoulder (not towards us) and accelerating toward us, steering the car towards the right. About 20 feet out we sounded our horn as we realized that driver wasn't slowing down (est their speed at 25 mph) and was steering into us vs. going straight past. Note that we were in the same stop location we would be had there been a stop sign for us.

The other car hit us hard, from a perpendicular angle... at their left front and our right front meeting. The impact pushed our vehicle 2 feet to our right and compressed/shifted the "nose" area in front of our front wheels about the same. The damage to our car was significant ($7K) as the sideways force bent the front of our vehicle from left to right, both frame rails bent, hood quarter panels, all engine components (AC, radiator, fans, generators) pushed to the right. Damage to the other car was relatively light, only their right front bumper severely damaged and a "scrape" down their right side as their vehicle continued past ours.

The other driver acknowledges "not looking" and we traded insurance info, called the police. No pictures, no witnesses (empty parking lot). Police report acknowledges we were stopped. Now the other driver claims we pulled out in front of her and she wasn't able to stop/avoid us and her insurance company GEICO has denied our claim. I would understand the "no fault" if we just "pulled in" to the lot and it was a "surprise" moment that caught both drivers off guard. However, having watched the other driver approach (without looking towards their direction of travel) for approximately 100 feet, and us honking our horn and seeing them steered into us (straight ahead would have missed us, much less if they would have turned one/two feet back to their left) I really question why the GEICO driver isn't being held liable for this accident. What recourse do I have to force the issue?

This is classic "he said - she said" case. Obviously the other driver gave a different accounting of what happened to her insurance company. So now her insurance company doesn't want to pay out. That's expected - given no police report that lays blame how does Geico know whether to believe you or their client?

My suggestion is to use your own insurance company to pursue this. If they can't get satisfaction for you, then you can always sue the other driver for damages. You have a witness (I assume since you used the word "we" in your post), so that's in your favor.

One other thought - did you or the police take any pictures? It sounds like the dispute is over precisely where your car was when she hit it - if you had pulled out in front of her then the accident would have occurred in the middle of the "T" intersection. If you didn't pull out in front of her, then the accident occurred before the intersection. So if you have pictures that show this, or if the police report describes precisely where the accident occurred in relation to the "T," you have good supporting evidence.

Definitely get your insurance company involved in this. That's what you pay premiums for. An agent will be assigned and will untangle this mess.