The ex-Manchester United star has come under pressure after their 1-0 defeat at home to New York Red Bulls on Wednesday.
That result has left them bottom of the Eastern Conference with just 15 points and two wins from their last 13 matches.
As well as unveiling a banner that read: "Neville out", fans let out their frustrations on his son, Harvey.
The 20-year-old, who players under Neville at Inter Miami, came off the bench for the final 10 minutes of the game but was jeered by supporters as he came off the bench before being booed each time he touched the ball.
But speaking post-match, Neville was having none of it, saying: "In moments of real adversity, you learn a lot about yourself and the players you have, and I was incredibly proud of the young boy they were booing.
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"It showed great courage. It hurt. It hurt. But I understand, and I think he understands. Do I agree with it? Probably not. But I'm his coach and I believe in young players."
Neville then directed a message to the Inter Miami fans, saying: "Hit me guys, I can handle that. I can take that. I've had it all my career.
"We have the best fans. I understand their frustration, the banners and all that because it comes with the territory being a professional football manager.
"I would say to the supporters, 'Just come for me.' Support the team and the young boys…Don't knock one of our own. Just come for me.
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"If you're unhappy, hit the manager. Don't go for one of my young kids. That hurts and that's personal."