Birighitti, who was among the Jets' best on Saturday night, had bravely fielded a through ball in his own box when he was collected by a racing Smeltz midway through the second half.
It was a collision not too dissimilar to when the ex-Gold Coast United striker left another former Jets gloveman, Neil Young, with five fractures in his nose in 2010.
Miller labelled the tackle "outrageous", "disrespectful" and "reckless".
"If you look at his mouth, for me, that's not on," Miller said after the match.
"Irrespective if its a mistake or unintentional, it's not right.
"(It) is not only disrespectful, but slightly out of order, if someone requires surgery post a one-on-one with a goalkeeper."
Miller stopped short of singling out Smeltz for the challenge.
"I'm not blaming anyone," he said.
"There's a border-line competitive side of the football match, and then there's outrageous and reckless."
Sydney FC coach Graham Arnold said his veteran striker cut a distraught figure in the dressing rooms at full-time.
"He said to me after it happened his head was gone. He wished we'd had another sub so he could've come off," he said.
"It's not good for it to happen to anyone. Let's hope (Birighitti) recovers quickly and well."
Miller was also left to rue an offside call that overturned what would've been a go-ahead goal from Jason Hoffman off a set piece in the first half.
Replays proved the Jets defender was offside by the barest of margins.
But the rookie coach remained positive with his team's overall improvement, including last week's upset of Wellington.
"Three points from a maximum of six, disappointing to lose at home, but we must make steps," he said.
"This is our progress."
In the first match of the post-Nathan Tinkler era, a healthy crowd of 12,173 greeted the new-look Jets, and Miller believed the locals would've been satisfied with their team's performance.
"They'll make their own minds up, but I'd like to think they will (come back)," he said.
- AAP