Discussing the Government's "get boosted by the bells" target on the BBC's Breakfast programme, he said: "We're at 72-point something before Christmas Eve.
"We've got appointments to get to 80%.
"What we need now is that final push to get people to come to fill those appointments."
He added: "That honestly is the best present you could give to any of your relatives, to get yourself protected so that you then protect others."
Prof Leitch also said he was "very, very hopeful" schools would return on time in January.
He spoke as new restrictions came into force in Scotland in a bid to tackle the Omicron variant.
From Monday, nightclubs will be shuttered and hospitality businesses will need to return to offering only table service if serving alcohol.
Bars, restaurants and indoor leisure facilities such as gyms, theatres and museums will also have to reinstate one-metre social distancing regulations.
The restrictions come after caps were placed on large events from Sunday.
First Minister Nicola Sturgeon announced last week that just 100 people would be able to attend a standing indoor event and 200 seated.
Outdoor events will also be restricted to 500 - a rule which has hit football games hardest.
Scottish football authorities took the decision to bring forward the planned winter break, meaning just one game will be played with the reduced number of fans.
The restrictions will be reviewed every three weeks, according to the Scottish Government.
Finance Secretary Kate Forbes said: "These temporary restrictions are targeted at reducing the spread of Omicron while the accelerated vaccination programme takes effect.
"I understand how difficult this has been for businesses in recent weeks but we must reduce our contacts and limit the spread of the virus.
"We're providing £375 million in business support, which is a significant investment, to support those who have experienced cancellations and help them get back on their feet in the new year.
"The best way to support business sustainably is to get the virus back under control. Please get your boosters and stay at home as much as possible just now."
The funding proposed by the Scottish Government will be covered using repurposed health spending sent north by the UK Government, as well as taking money from commitments already made, the Finance Secretary said in a letter to a Holyrood committee.
Further spending could be taken from next year's budget, resulting in a fiscal shortfall if the UK Government does not pledge the gap.
The last available figures released by the Scottish Government show 7,076 new cases of Covid-19 reported on Christmas Eve, with 10 deaths and a positivity rate of 14.6%.
A total of 6,154 cases of the new variant were also reported on Friday, up 3,832 from the previous day, due to a backlog in laboratory testing.