In the encounter in a shop, the care worker told him she was "very passionate" about her job but claimed the disability sector was "a joke".
"You've done nothing for us, our people are suffering," she told him.
Harris told her it was "not true" for her to say that carers had been ignored in the budget and insisted he was also very passionate about disability.
He tried to bring the conversation to a close by shaking her hand and walking away.
"Keep shaking hands and pretend you're a good man," Ms Fallon replied, clearly annoyed by how her questions had been received.
"You're not a good man because you don't care about our people," she added.
The following morning, Harris phoned Ms Fallon to apologise to her and to discuss her concerns in detail.
Speaking later on Instagram Live, he said he "didn't give her the time that I should've given her and I feel really bad about that because it's not who I am".
The 38-year-old politician became Ireland's youngest ever taoiseach (prime minister) in April.
His involvement in politics began at the age of 16, when he began campaigning for better support services for his autistic brother in his home town of Greystones.
Harris added that if he gets a chance to serve as taoiseach again he is determined to prioritise the issue of disability.