How fast does a car have to travel, to spin if hit?

I recently was in a car accident. My side of the story: I stopped at a stop sign traveling west and proceeded into the intersection. I estimate that I could only have been traveling between 5-7 miles an hour.
I made it about 1/2 of the way into the intersection when the other vehicle, traveling south sped through. I never say the vehicle. I hit the back 1/4 panel of his vehicle causing it to do a 180 spin and stop. There were no skid marks from either vehicle. My vehicle was a 2013 Ford Escape, his was a 2013 Dodge Avenger. My vehicle was totaled, he had less than $7,000 damage to his back end. My question is, how fast was he traveling to spin like he did? I have an idea from watching 3 other cars whiz by but I could use some 'expert' opinion.

Sorry but it would require a professional; accident analyst and a lot of complex math to perform the physics calculations needed to answer this question. The analyst would need to know the ambient temperature, the pavement temperature, any degree of incline or crowning of the pavement, the coefficient of friction of the tires of the vehicle that spun as well as the coefficient of friction of the section of pavement where the collision occurred, the exact loaded weight of each vehicle plus the exact paths of both vehicles and the exact point of impact on each vehicle.

I understand it would be difficult. I guess I didn't realize how difficult. Can you tell me if it is possible?

Zero. A car doesn't have to be traveling to spin. Even a parked car can spin if hit (although normally not if hit by a vehicle going as slow as you say you were going). I've even gotten a stopped car to spin without being hit (it was very icy, so when the front of the car stopped for a traffic light, the back kept moving).

You seem to be insinuating that he was travelling at a high rate of speed. You seem to be basing his speed on whether or not the car spun. What difference does it make now to know how fast he was travelling? If it is really important, you could hire a accident reconstruction expert. He may be able to tell you. Or you can hire a physicist.
All that aside, it sounds as if you were cited for failure to yield. If I'm correct then it doesn't matter how fast he was going. Because we're not allowed to proceed if the way is not clear. He could have been going 100 mph and you would still be found liable and guilty of failure to yield.

If I had to guess, you was cited for failure to yield, or that the insurance is stating that you "appear" to be at fault for this accident?

I was told years ago, that it is rare for skid marks, since most cars have antilock brakes? As for the spin like others stated does not have to be because you "think" the other driver was driving at an excessive speed. If his car was hit in the 1/4 panel, and at the (right) angle could have cause it to spin, vs hitting on the side? Now did you hit head on or try to turn/avoid hitting it? How many lanes are in this intersection? Was it 2, 3 or 4 lanes when you met in he middle? Did he have a stop sign also? Was is the speed limit? Are there obstructions to not allow you to see (clearly) any traffic approaching the north or the south?

(IF) both of you had stop signs, then my guess is that both of you claim to stop and neither one of you saw each other, or that both of you did (not) stop, but could have slowed down to "coast" into the intersection.

The insurance company for both of you will get recorded statements of (how) the accident happened, then also review the police report, (plus) they will look at (both) vehicle damages to make a final decision of (who) is at fault, or they could find that (both) of you are at fault for this accident. The insurance adjusters and the auto appraisers and (sometimes) they might have an accident expert for their opinion. It will be up to them to decide.

Depending on exactly where a car is hit, and the tyre/road conditions, it is possible for a car that was stationary to be made to spin.

Excellent physics question, too bad I can't answer it

" I hit the back 1/4 panel of his vehicle causing it to do a 180 spin and stop."
so 5 to 7MPH as YOU hit him.
I would think that was obvious.
You had the STOP SIGN. Not him. So you HAD TO WAIT until the road was clear. YOU DIDN'T.
NEXT!

A stationary car can spin if hit right. So your claculations are meaningless.

I'm guessing you'd like to, at least partially, blame this accident on the other driver's excessive speed. That's not going to happen, not even if your lawyer is Johnnie Cochran's ghost. Anyway…

It doesn't require much speed to make a car spinout, and I'm not talking about winter roads here. If a vehicle is travelling at 30 km/h or faster, any sudden change to the vehicle's direction will easily cause the tires to lose traction. Any change at all, such as a sudden turn or (just for example) being struck by a slow-moving Ford Escape.

If his vehicle spun 180 degrees, my closest guess would be he was doing 50-65 km/h when it happened. Any faster and the impact would have sent large pieces of both vehicles flying.