Darren Ambrose's 40th minute free-kick gave Palace a lead that was cancelled out by David McGoldrick's leveller three minutes after the break.
But Forest manager Davies was enraged after Hill appeared to elbow Anderson on the stroke of half-time, sparking a melee in front of the dug-outs.
Davies leapt to his feet - and the atmosphere became even more charged when the Crystal Palace radio commentary was fed through the stadium public address system just as the commentator described the visiting manager as being "out of his pram".
Hill and Forest right-back Chris Gunter were booked for squaring up to each other during the skirmish.
But the Palace man escaped further punishment after referee Graham Horwood decided the challenge was not deliberate.
Davies said: "I would say it was a red card. And when Neil Warnock doesn't complain to you at the side of the pitch then you know he also knows.
"He's a very fortunate manager and he's a very fortunate player because it was a very blatant elbow in my opinion.
"The referee says there was no malicious intent which I find quite staggering.
"Paul Anderson is okay, but he's a lucky boy. The medical people wanted him to come off straight away and he didn't look too comfortable.
"I was out of my pram because I'm trying to protect my player, 21 years of age who had just been hit in the head. And I'm really sorry if I've upset this tannoy guy. Tell him I'm sorry!"
And he added: "Neil's very good at making it look the opposite of what's happened. He's a great lad, wears his heart on his sleeve and I've great respect for him and what he has done. But sometimes he's on a different planet."
But Palace manager Warnock believes the referee got it right and was furious to discover Davies had described his side as physical and hard to handle.
Warnock said: "I think that's a disgraceful statement. If I'd spent nearly £4.5million on players, I'd probably say a bit less than that.
"He should keep his mouth shut and as how come we carved them up so often and created so many chances.
"I thought it was a yellow card, not a red card. It was a reckless challenge but it wasn't deliberate."
Ambrose had already tested Forest keeper Lee Camp with a fourth minute free-kick when he found the top corner five minutes before the break.
But McGoldrick's close-range finish from Lewis McGugan's low cross brought the visitors back into it, although Davies' side could not find the goal that would have brought a sixth consecutive victory.
Instead it was Palace who had the better chances to claim all three points with substitute Victor Moses somehow spooning over from just six yards.
And Camp produced an outstanding save to deny Ambrose a second goal with a point-blank volley.