Have you tried any counter conditioning or desensitization training? If not, those are viable options here.
You'll want to first identify when your dog begins to feel anxious. Is it when you start driving, when you grab your car keys, when you turn the car on, or at some other point? Once you identify that moment, go back one step and start training there. So, if your dog starts to feel/act anxious when you start the car you'd want to begin training with the car off or maybe even outside the car to be safe. For this example, I'll assume your dog is fine until he gets into the car and give you a basic protocol to follow from there.
- Get a bunch of high value treats (stuff like cheese or fresh cooked meat, these treats should basically be the best thing your dog can imagine).
- Go through the motions as though you're going to take him for a car ride - put your shoes on, grab your keys, etc. and start to walk outside.
- As you're walking towards the car, give him a treat, then another, and another one. Also tell him he's a good boy and pet him.
- Go right up to the car and again treat and praise. If he looks anxious at all or at any point stops taking the treats then stop moving towards the car and go do something else.
The point here is to not push your dog over the edge. You want the training to be a positive experience where he comes away thinking that car equals treats and petting.
Rinse and repeat until your dog no longer shows any signs of anxiety when going up to the car. Once you have had a few successful times of no anxiety near the car you can open the door. Don't ask him to get in, but treat and praise like crazy if he does.
Take small baby steps in this training. Open the car door, then have him get in with the door open, then move on to closing the door, then turning the car on. All the while if he ever seems anxious at any point back up a step in the training.
While you don't need to medicate your dog, the training might go quicker if you're able to either avoid bringing him in the car for the near future (until he's not as anxious anymore) or you're able to artificially (with medication) calm him down during car rides. Again, not necessary but may make the process go quicker. Of course, I don't know your dog so you should talk about any medication with your vet and if you feel being in over your head there's no shame in talking to a local dog trainer experienced in positive reinforcement training.
If your dog gets anxious before he even sees the car, like if he's anxious when you pick your keys up, you can use a similar protocol. Using the keys example, you'll want to pick your keys up randomly throughout the day, but then don't go to the car. Pick up your keys and then go sit on the couch and read, or pick up your keys and just move them to a table or something. Basically you want to break the association of keys being equal to something bad being about to happen and teach your dog that picking up your keys is just a weird thing humans do sometimes and it's nothing to worry about.