No, he is the son of Prince Charles:
According to this news source, this was confirmed by a blood test:
Diana, who publicly admitted to her affair with Hewitt in a 1995 television interview, insisted that the dates of her affair with the officer meant he could not possibly be Harry's father, Simmons said.
However, she was pressured to carry out DNA tests on both Harry and his elder brother, Prince William, to prove their paternity, the Sun said in its Wednesday edition.
The tests showed that both princes had been fathered by Charles, the heir to the British throne
—source
According to James Hewitt it is impossible because he started his relationship with Princess Diana after Harry was born:
"There really is no possibility whatsoever that I am Harry's father.
Although I was with Diana for a long time I must state once and for all that I'm not Harry's father
"I can absolutely assure you that I am not."
Prince Harry was "already walking" by the time his relationship with Diana began
—source
This was also confirmed by Diana's body guard:
In a book by Diana's former police bodyguard Ken Wharfe, the rumours are also quashed.
Mr Wharfe said the gossip used to upset the Princess.
He said in his book: "The malicious rumours that still persist about the paternity of Prince Harry used to anger Diana greatly.
"The nonsense should be scotched here and now.
"Harry was born on 15 September, 1984.
"Diana did not meet James until the summer of 1986, and the red hair, gossips so love to cite as proof is, of course, a Spencer trait."
—source
