Is Inspector Spacetime a reference to a series that really exists?

Inspector Spacetime is a direct parody of the series Doctor Who.

Some clear parallels:

• Inspector Spacetime travels in a time traveling phone box:

  

  The Doctor travels in a time traveling police box:

  

• The enemies of the Inspector are know as the Blorgons:

  

  Which are a parody of the Daleks from Doctor Who:

  

In addition to onewho's excellent answer, I'd like to point out how Inspector Spacetime was initially introduced into the show.

In 3x01 Biology 101, Britta gets Abed interested in a British TV show called Cougarton Abbey (because of course Britta only watches European TV), which is an obvious parody of the British series Downton Abbey (and a wink to the US series Cougar Town, which, if nothing else, you should remember from the My Diner With Abed episode).

The one thing that is important to know about real-world UK series is that they are usually quite short by American standards. The original The Office, for instance, had just 2 seasons of 6 episodes each. So Abed is devastated when he finds out that Cougarton Abbey ends after only 6 episodes, with all the main characters dying.

To cheer him up, Britta tells him of a popular British sci-fi series that has been running for decades, Inspector Spacetime (Doctor Who holds the Guinness world record for longest running sci-fi show).

This keeps Abed happy in several ways as it appeals to his new-found interest in British TV, as well as his existing passion for sci-fi, without the risk of running out of episodes to watch anytime soon.

Inspector Spacetime is a direct parody of Doctor Who.

Inspector Spacetime is a direct homage to the long-running BBC science fiction TV show Doctor Who. The show within the Community universe follows closely the story of Doctor Who with the Inspector and his partner traveling in a red telephone box known as the X7 Dimensionizer time booth, the counterpart to the T.A.R.D.I.S. - and using his Quantum Spanner to fight alien creatures such as the Blorgons.

Source: Community Wikia