The new restrictions – banning gatherings of more than six people indoors and outdoors – came into force on Monday.
It is intended to simplify and strengthen the rules on social gatherings amid rising coronavirus cases .
Mr Buckland’s denial follows criticism from a source close to the Archbishop of Canterbury about the social restrictions imposed this week to prevent the spread of coronavirus.
Speaking to BBC Radio 4’s Today programme, the Cabinet minister said: “Archbishop Justin (Welby) makes an important contribution to this debate and he is right to point to the huge spiritual and social significance of Christmas.
“I don’t think any of us in Government want to be Oliver Cromwell-esque about this – we want to see families celebrate Christmas in a safe and happy way and we want to see our churches and indeed our other places of worship joining in that celebration.”
Mr Buckland added: “We are not going to cancel Christmas but the ‘rule of six’ is something that is clear and important and I do think we’ve committed to that and we need to stick to it.”
However, Mr Buckland admitted that the Government was facing “real challenges” on coronavirus testing and suggested that school children and their parents would be the next testing priority after NHS and social care workers.
It comes as people opted to turn up to hospital A&Es on Tuesday in a bid to get a Covid-19 test due to a lack of available bookings through the online system.
Mr Buckland told Sky News: “I think laboratory capacity has been an issue, we’re working our way through that, we’re increasing the number of test centres – we’ve got 400 test centres, getting it up to 500 – but clearly there are still real challenges.
“I think the announcement by (Health Secretary) Matt Hancock yesterday to create a prioritisation system is the right thing to do.
“He is going to develop that very quickly over the next few days, to explain to us what that looks like but I think… it has to be the NHS first and then social care.
“And then I think what we need to do is have a cascading system where we know where our priority should be and for me priority should be for children in school and their parents in order to ensure their lives are safe and also importantly they are not disrupted in the way we are seeing.”
During his media rounds this morning Mr Buckland was adamant that testing capacity was “ramping up” to deal with the demand.
Speaking to Sky News, he said: “I’m not shying away from the current issue but what I’m trying to explain is that rather than us sitting back and pretending all is well, we have accepted the scale of the challenge, we’re ramping up the test centres, we have increased laboratory capacity, new labs coming on-stream so we can get that quick turnaround.”
He added: “The fact the Government kept on saying about the dangers of a second wave, at all times the Prime Minister, all of us, were absolutely focused on the dangers of the second wave – we have seen what’s happening in France.
“We absolutely are onto this in terms of understanding that through the autumn, if we are to get the balance between getting the economy back on track and getting children into school, then all of us now have a special responsibility to follow all those guidelines and do whatever it takes to beat this virus.”