I had a car accident in TX. The insurance company of the fault party is passing the wrecker service responsibility to me accusing me for not mitigating the damage after the accident. However, the fact is that they have been delaying my case. Three weeks after the accident, they were still waiting for the police report and a statement from their insured. I purchased only liability coverage. Therefore, my insurance company is not responsible for fixing my car. I did not fix my car because I have been waiting for the decision of the faulty party's
Insurance company regarding if they are going to repair or to total my car.

How can I make the fault party insurance company to pay for the wrecker service? Thank you

Assuming things go well it will likely take several months for everything to be settled, so be patient. But please clarify what you mean by "accusing me for not mitigating the damage after the accident." Are you saying that the reason you needed towing service is because of something that happened AFTER the initial accident? Also, do you have a copy of the police report, and does it specifically lay fault on the other party?

The only way you can "make" the insurer pay is to sue them. I'm also curious about the insurer's claim.

Often they delay and delay to try and get you to accept a lower settlement. And they will normally try to deny and pay less if possible.

You may have to sue the other driver (or/or car owner)