Speaking to Gary Robertson on Good Morning Scotland on Tuesday, Mr Leitch described delays in the testing system against a background of 8,252 positive results on December 25, 11,030 on December 26 and 10,562 on December 27.
It has been reported that some people in Glasgow have had to wait four days, or 96 hours, for a PCR test result.
The usual wait is 24 hours and can be less depending on the time at which the PCR test is taken.
Asked why people had had to wait such a long time for test results, Mr Leitch said it had just been scale.
“It has just been big because quite a lot of people went for testing before Christmas, in order to try and get Christmas celebrations that were safe if they had symptoms or it they were contacted,” he said.
“I apologise for that. I am sorry if people had to wait a little bit longer.
“We think the acute problem, the immediate problem is resolved.
“My understanding was, yesterday (Monday) speaking to test and protect leaders, that the backlog was cleared and they were back to somewhere between 24 and 36 hours for pretty much all the tests.
“Postal ones take a bit longer just because it takes a while to get through the mail and back.
“Fundamentally, yes, if there is a massive increase in PCR testing, then you do have to make some choices about prioritisation, about high-risk individuals and locations, but we are not at that stage yet.
“We need to wait to see the numbers smooth out a little, to give the non-statistical phrase, just because Christmas testing behaviours are different, but if you look back to the day when Omicron first reared its head in Scotland, we had one case on November 28, we had 2,332 other cases, pretty much all Delta.
“Now we have four times that and it’s three-and-a-bit weeks later, nearly four weeks later, so, it has quadrupled in three weeks which is exactly what we said would happen, I am afraid.”
Speaking about the numbers in hospital, Mr Robertson said that up until Christmas Eve, 42 people with Omicron had been admitted to hospital, which on the face of it seemed very small.
Mr Leitch replied: “We have got used to these numbers, haven’t we.
“Before the pandemic our baseline intensive care beds for the whole country was about 200.
“So that’s two fifths of our intensive care base being made up by Covid cases, but you are right, that is down on our high numbers of well over 100 at some of these waves.
“The hospitalisation numbers are about the same proportions, somewhere about 500 or so, and have been declining with Delta.
“They are not rising at the rate they rose with Delta, which is excellent news and confirms, maybe, the data we saw a few days ago before Christmas about this being a milder disease, but milder does not mean mild.
“We should not overreact, we should absolutely not, but nor, and you would expect me to say this, should we underreact.”