The messages between the minister and former NSW premier Gladys Berejiklian were revealed by Network Ten, and labelled the prime minister a "horrible, horrible person" who was "actively spreading lies".
Barnaby Joyce says the messages should be ignored because of the leaker's intention to deliberately damage the government.
"You are doing this for the purpose of malice, for the purpose of vindictiveness and for your own personal grudge and therefore, it is to be disregarded," he told the ABC.
"It's not so much what is said, it's the motivation. The Australian people aren't stupid. They'll say, 'Yeah, I could show you texts between brothers and sisters but we never expect you to show them to mum or dad'."
Mr Joyce also labelled former Liberal prime minister Malcolm Turnbull "an angry man" who finds reasons to attack the government.
Mr Turnbull said he wasn't surprised by the content of the messages, only that they made their way into the public domain, and that the government is in "plenty of trouble" coming in eight points behind Labor in the latest Newspoll.
Defence Minister Peter Dutton said people say things that are exaggerated and heated in the moment, and the messages didn't represent Mr Morrison's character.
"(Putting) the petty discussions to one side, most Australians see they've got a prime minister who is human like the rest of us, doesn't get everything right, but dedicates himself each day to get us through the realities of COVID," he told the ABC.
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