The Good Morning Britain host said he had reported the threat, which was made on Instagram, to police and was also working with Instagram owner Facebook to track the person down and hold them to account.
Last week, the unknown person left a comment on Morgan's eldest son Spencer's Instagram account that included the words: "This isn't a threat, Piers it's a promise, you're getting killed."
Addressing the incident on Monday's Good Morning Britain, Morgan said: "Let's see how bold and brave they are when we actually track them down, which is going to happen.
"As they said in Taken, 'We will find you.'"
The presenter said both the police and Facebook were investigating what had happened and promised not to stop until the person had been caught.
He said: "I'm going to push this through right to the end. I'm going to find out who did that threat and they're going to be held to account, and they'll go to court if necessary, and I will appear in court if necessary.
"That's what I'm going to keep doing every time one of these morons thinks that's OK, because it's not OK."
Morgan added: "I'm going to go through the process and see what happens because I want to know, can you find these people? They've had their account deleted, OK, but who is this person who felt that it was acceptable to issue a death threat on Instagram to my son?
"My son's fine about it, he's been my son a long time, and he's used to the rough and tumble of public life vicariously through me. However, it's just a line crossed, and it's got to stop."
His co-host Susanna Reid encouraged him to pursue the person on behalf of everyone who had been trolled.
She said: "I think it's important for you to make sure that process is exhausted, not just for you, but also for people who don't have your sort of influence."
Morgan agreed: "You can't make a death threat to someone in the street, you can't go up to someone and say I'm going to kill you, I'm definitely going to kill you, because that is a criminal offence, but you can do it on social media, so I'm going to do it for all the people who are getting these sorts of threats and have no recourse to do it."
In the same show, Morgan, who is no stranger to social media spats, also urged anyone "spewing vicious abuse" about people including Marcus Rashford, the late Captain Tom Moore and Prince Charles to "just shut up".
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