The answer to your question is no.

There have been questions about Edward IV's birth since his reign. His brother Richard III even declared that Edward's children (the Princes in the Tower) were illegitimate (obviously, though, this was to take power for himself).

It began with Edward's birth in Rouen, away from England, and the hurried baptism in a side chapel (whereas their second son, Richard, was baptised in a large and lavish ceremony). Other evidence used to support the case of illegitimacy were records found about a decade ago in Rouen Cathedral that indicated that Richard of York was on campaign at Pontoise from July 14-August 21 1441. As Edward was born April 28 1442 this calls into question the timeline of his birth. With that said, it would not have been uncommon for Edward to have been born premature (and that would fit with the hurried baptism). Proponents also argue that Edward's appearance was nothing like his father's.

It was common for prominent political figures of the time to have their legitimacy questioned. No direct evidence exists that indicates Richard of York wasn't Edward IV's biological father. Also of note was that Richard of York never gave any indication that Edward was not his son. At the time, allowing an illegitimate child to be the heir to the house/faction was aiken to treason so if Richard even had suspicions it would be his right and duty to challenge the birth and he did not.

However, as it turns out, all of this is moot. What the program you watched probably failed to discuss was that Edward's claim to the English throne was legitimate through the Right of Conquest. When he defeated Henry VI he had every right to claim the crown from this act, independent of his birth or his status. This was one of the strongest contemporary cases in favor of the Yorkish claim to the throne, as well. In the order of strength of possession Right of Conquest is the strongest case for the throne, followed by Presumption and the Right of Royal Succession.

Also of note was the fact that Edward could reasonably claim a direct blood line to Edward III through his mother Cecily who was the grand-daughter of John of Gaunt. This is the same claim that Henry Tudor eventually used in 1485 to claim the throne. Technically this line was barred from the throne because John of Gaunt and Katherine Swynford were not married at the time of Cecily's mother's conception. However, John of Gaunt later married Katherine, which legitimised her issue and Richard II later confirmed this legitimacy, with the caveat that the line would not be eligible for the throne.

tl;dr: In the end, the question of Edward IV as illegitimate is inconclusive and largely irrelevant. There is no direct eveidence for it. But even if it were the case he still had a claim to the crown through the Right of Conquest.