Are there road bike tires invulnerable to popping?
At risk of being downvoted, you don't want want solid tires. There's a reason that all performance tires for all vehicles are pneumatic. The decrease in compliance of a solid foam or rubber tire just doesn't compare to a pneumatic tire.
Look into a tire with kevlar reinforcement- most brands have them- or aftermarket anti-puncture tire liners that sit between the tire casing and your tube. You may even want to try a tire sealant like the one made by Stan's NoTubes, though if you're running tubes you'll need the kind with removable valve cores.
One last thing to note is that the number of flats you're getting may depend on what part of the road you're riding on. If you're riding to the right of the white line (assuming you're in the US), ie on the shoulder then you're going to get a lot more flats. The shoulder is where all the debris accumulates even if the road looks clean. For many road riders, especially those who are just starting and don't feel comfortable riding on the road, simply switching from riding on the shoulder to riding to the outside of the actual lane alleviates a substantial portion of flat issues.