Use of "on" in this sentence

Right, the on heads a prepositional phrase and identifies that the information following will be a date. See Merriam-Webster, "on," preposition entry 3:

—used as a function word to indicate a time frame during which something takes place or an instant, action, or occurrence when something begins or is done

If the sentence already identifies the subject as a date, then there is no need for a preposition to clarify that you're talking about the date.

Compare:

The date was July 11, 2019.

The inspection was on July 11, 2019.

That said, there are exceptions. On clarifies the relation between its object and the rest of the sentence. "The date was on July 11, 2019" is redundant, but appears sometimes in published work:

The inspection date was on September 24th. ("Skimmers found at Bradenton gas station")

Similarly, one can omit the "on" when date is missing, but it may sound less formal because you're leaving more interpretation to context. This tends to happen in speech:

The inspection was August 24th, 1927. (New York Court of Appeals, verbal transcript)

I think you are correct:

The most recent inspection of the building was on July 11, 2019.

or:

The most recent inspection date of the building was July 11, 2019.

or:

The most recent inspection date was July 11, 2019.

Would be better.

The most recent inspection date of the building was on July 11, 2019 is a bit of a tautology; "inspection date" and "on July 11..." are dealing with the same thing but in a different way.

• Date assumes knowledge of the actual date you are speaking about.
• The third example assumes knowledge of the thing that was inspected...