No, damage done to a car due to putting diesel into an unleaded gas tank normally is not covered under physical damage coverage.
Accidentally putting in the wrong type of fuel can cause mechanical damage which is not usually covered under comprehensive (other than collision) coverage.
You mentioned that you had $5000 worth of damage due to using the wrong type of fuel; you may want to check with a few mechanics to find out exactly what you will need repaired in this situation. The repairs may not be that extensive, especially if you did not drive at all or that far before realizing the mistake.
Filling a gasoline engine with diesel is not as bad as using gasoline in a diesel engine. Many times there is no permanent damage if the mistake is caught soon enough.
The usual advice given by a mechanic to someone that has put diesel into a car that uses unleaded fuel is to not start the engine but tow the car home and get as much of the diesel fuel out as you can by siphon or any method available to you. Once the diesel has been removed put in your correct unleaded gasoline. While it may be difficult to start it usually will and once started it will smoke a bit until the diesel residue works through the system.
Or you may want to have the fuel tank drained of diesel fuel drained by a professional and have the whole fuel system flushed out since if the diesel stays in there too long it can ruin the injectors, fuel pump, fuel lines, etc. If the diesel gummed up the spark plugs and filters, they will probably need to be replaced.
Gasoline engines use spark plugs to ignite the gasoline vapor. Diesel engines compress the liquid fuel so that they reach a peak temperature and pressure for the fuel to self-ignite. Gasoline performance is measured in octanes whereas Diesel performance is measured in cetanes. Diesels can self-combust while gasoline engines are not designed for self combustion.
The core of the problem with mixing gasoline and diesel fuel is that in a diesel engine you want controlled self-ignition, whereas in gasoline you want to avoid self-ignition, and have the fuel ignite only when the spark fires.
My mechanic suggested if I were to put diesel fuel in my unleaded vehicle to remove as much diesel fuel as possible from the gas tank and refill it with gasoline. He said I may have to turn over or crank the car to purge the diesel fuel out of the fuel lines. After that it would likely run okay but as mentioned above if the car was run on the diesel fuel the fuel filter, etc may need to be changed out.
So depending on the proportion of diesel fuel relative to gas in the tank, a gasoline engine will normally either run poorly or stop altogether if you have used diesel fuel in it. If it stops then a draining and flushing of the fuel system can be needed. The suggested cost of this from the sources I have seen say it should cost only around $500, so you may want to speak with other mechanics or repair shops if you have an estimate of $5000 worth of repairs needed.