After boiling pasta and setting the water in which the pasta was boiled aside I noticed that after the water cooled it thickened. It sort of has the consistency of gravy. Would I be able to make gravy with this instead of using starch?
No you would not. If you are cooking pasta properly, you need to use a lot of salt, rendering any gravy made with this water far too salty.
In addition, gravy is properly made with stock; made with water it will have no flavour.
You can't make a 'true' gravy, but you can put a little of the water into the sauce you are using to loosen it a little if it has reduced too far, or to make it go a bit further.
I think you should read McGee on this one. Basically he says the 'water' from boiling pasta is very rich in flavor.
Italian recipes often suggest adding pasta water to adjust the consistency of a sauce, but this thick water is almost a sauce in itself. When I anointed a batch of spaghetti with olive oil and then tossed it with a couple of ladles-full, the oil dispersed into tiny droplets in the liquid, and the oily coating became an especially creamy one.
Restaurant cooks prize thick pasta water. In “Heat,” his best-selling account of working in Mario Batali’s restaurant Babbo, Bill Buford describes how in the course of an evening, water in the pasta cooker goes from clear to cloudy to muddy, a stage that is “yucky-sounding but wonderful,” because the water “behaves like a sauce thickener, binding the elements and flavoring the pasta with the flavor of itself.”
Just to add a little to this discussion. There should never be that much starch left over in your water to really thicken a tomato sauce. I will have to back @ElendilTheTall here. All the thickening power of the starch should already be completely used up in that cooking water, and the cooking water should be thinner than your tomato sauce so the net result should be thinning it.
That being said something that has been thickened with starch will as it cools get thicker (its actually more of a gelling or coagulation). It should actually thin back out if you reheat it, albeit maybe not to the point it once was. But that might be why you noticed your pasta water getting thicker.
Most Italian chefs that I know add pasta water to there sauce to bring it together with the noodle as starch is attracted to starch. The principle being it will help the sauce better coat the noodle.
If you want to get clever though, I can definitely say you can make gravy with pasta water. Many pasta sauces are actually referred to as gravy. So could say the finishing touch to your gravy would in fact be pasta water :)
As for a thanksgiving style gravy I would say I hope not. If your pasta water has enough residual starch to thicken up your stock and/or pan drippings then you probably aren't using enough water when cooking your pasta.
I am ethnically Italian. Whenever my grandmother would make pasta she would keep the water that was leftover. Within a day or two she would combine the water with Parmigiano rinds, whole onions (skin and all), celery ends and carrot knobs (all cooked very slowly) to make the broth she would use for vegetable soup, pasta sauce and the liquid to raise the browns from cooked meats (veal particularly). The broth left-over is so rich and delicious with a flavour incomparable to anything I've experienced in other cuisine. KEEP IT. USE IT.