No, Peter Parker was never diagnosed with HIV or any other serious illness. That's not what the show in question was trying to say.
A retrovirus is a very huge classification of viruses that work somewhat differently that other viruses. When a retrovirus infects a cell, it takes the virus's RNA and splices it into the existing DNA of the cell; from there, the cell will automatically start reproducing the virus as part of it's normal life cycle.
Retroviruses are used in medical research, especially in genetic engineering, because they allow us to modify the DNA of an existing, adult organism in-place.
The show you watched was merely explaining that this is one possible way that Peter could have obtained his power: a retrovirus infected his body and invaded all of his cells, injecting new DNA into Peter's existing DNA and giving him his powers.
It had nothing to do with any form of fatal disease.