I had this problem the first time I used my tire inflator in my G6. I opened both the fuse box in the passenger compartment and under the hood and found no blown fuse. Later, I learned that there's an additional fusebox in the trunk. Opened it up and sure enough, there was another oddly labeled fuse that was blown and it was the one.
I have since learned that on all my cars, no matter what, running the inflator for more than 3-4 minutes blows the fuse, every single time. I have never used the tire inflator without it blowing the fuses. I'd imagine the things could damage an electrical system (in theory) but the fuses blow before the inflators draw the amount of power it would take to do so, so I'm not convinced it could have damaged your car in any other way more than blowing the fuse. Don't rule out the fuse unless you're absolutely certain you checked the correct one (they could be on a different fuse then you're expecting) and have tried replacing the fuse even if it doesn't look blown. The fuses are small and sometimes it's hard to visually verify they're blown. If you've exhausted the fuse, then the cigarette lighter probably needs replaced (maybe the adapter on the compressor bend the contacts.) If you have more than one cigarette lighters and both don't work, verify the fuse situation one more time, and if it still doesn't work, I'd take it in to get checked. Electrical system is nothing you want to mess with yourself in a Lexus and it should be covered under warranty if you still have one.