Well kinda, but don’t expect it to work very well. The first thing you need to know is that a satellite dish is an antenna, but it’s designed to pick up signals in a completely different band from a TV antenna. You won’t get a lot of signal just by trying to connect the dish to the TV’s antenna in port. In fact it would surprise me if you got anything at all.
But let’s say that you bought just the reflector and arm assembly and then attached a small antenna like the HD-BLADE mini to it. In a case like that I would expect to get some signal, but not very much. The reflector of the dish focuses all the signal into a very small area, so most of the antenna would be unused. If you got any signal it would be just from the antenna pointing the right direction.
I’ve seen dish-style antennas for TV reception and they work, but connecting a satellite dish to an antenna is just going to disappoint you. The idea is sound but it just doesn’t work. It’s possible that if someone spent enough money engineering the thing, it could possibly work, but the resulting antenna would probably just be too small and wouldn’t necessarily work better than a small outdoor antenna like the Xtreme Signal HDB2X. The horizontal bars on the HDB2X accomplish the same function as the reflector on the dish (the “dishy” part) as they focus signal onto the antenna elements, which are the X-shaped parts on the front. This is a much more economical way of dealing with the same problem and works very well for antenna signals. Of course, something like that isn’t going to do well for satellite signals, which is why satellite dishes are shaped the way they are.
There’s a lot of physics involved in explaining why different antennas are better for different signals, but it boils down to this: imagine an antenna is like a net, catching different signals. Some signals are physically larger than others (this is actually true) so you need a different size net to catch them while still letting other signals that you don’t want pass through, and using as little material as possible so as to save money. There’s a lot (a LOT!) more to it than that but that’s a good way to think about it.