The Forest manager was sent to the stands for taking his protests too far in the tunnel at half-time. The Scot had complained to referee Anthony Taylor after Leicester defender Wes Morgan stayed on the pitch having conceded a first-half penalty with a clumsy foul on Jamie Mackie, from which Andy Reid fired the home side into a 2-1 lead.
Leicester's Paul Konchesky was then shown a red card for a lunging tackle on Mackie in the 60th minute as Davies looked on from the directors' box, but substitute Riyad Mahrez fired the 10 men of Leicester level with eight minutes to go.
After seeing his side get themselves in a winning position against a team short on personnel, a frustrated Davies said: "Wes Morgan was the luckiest player on the pitch. It was a clear goalscoring opportunity.
"I had an exchange of words with the referee. I asked him why it was not a red card, he said that it was not a goalscoring opportunity. He sent me to the stand, for raising my voice."
Davies added: "I have to say I think that the players gave us everything. Everything. We played against a Leicester side that was playing their strongest team and had 10 days of rest.
"We just ran out of juice. We were disappointed with the goals we gave away, but this was against a side at the top of the table and a side that has developed well over the years.
"These are little things that go against you. But the positives are there as well."
Sky Bet Championship leaders Leicester had gone in front through a Jamie Vardy strike just before the half hour, but Jamie Paterson cancelled that out and Reid's penalty gave the hosts an interval lead.
Leicester manager Nigel Pearson praised the fighting spirit of his side - although he refused to get drawn into any criticism of referee Taylor, who awarded a penalty that led to Leicester's equaliser.
"These things happen. Wes, in the first half, just fell on him (Mackie). There was nothing malicious. It was just a wee bit clumsy. That happens in games," he said.
"I try not to speak too much about individual incidents, the officials have a job to do and I can't make too many comments about it, without seeing things again."
Of the red card for Konchesky, he said: "It was a late challenge from Konchesky, I accept that.
"It probably looked a wee bit worse that it was, but I won't make too many more comments on it."