And Mr Trump said that he wants better terms for the US in trade with Europe, complaining that there was currently a "one-way street" for car exports - particularly from Germany.
Speaking to UK's The Times and German newspaper Bild, the president-elect said he did not care whether Washington had to deal with individual nation states or the EU as a whole - though he complained about the slow process of getting EU approval for a construction development in Ireland, accusing Brussels of using "environmental tricks" to block it.
He said that keeping the EU and the euro single currency together would not be "as easy as a lot of people think".
Mr Trump described European Commission president Jean-Claude Juncker, who rang him to congratulate him on his election victory, as "a very fine gentleman". But asked his view on the future of the EU and the likelihood of further countries voting to leave, he said: "I think it's tough ... I think people want their own identity, so if you ask me, I believe others will leave."
Trump described German Chancellor Angela Merkel as "a great, great leader", but said her decision to open the country's doors to Syrian refugees was "one very catastrophic mistake".
He restated his intention to enforce "strong borders" around the US, with early action to impose "extreme vetting" of Muslims.
Asked whether this could involve new restrictions on European nationals coming to the US, Mr Trump said: "Well, it could happen, I mean we're going to have to see.
"I mean, we're looking at parts of Europe - parts of the world and parts of Europe - where we have problems, where they come in and they're going to be causing problems. I don't want to have those problems."
Mr Trump said he felt "great love" for both the UK and Germany.
But he expressed frustration with terms of trade which he felt were weighted against the US, contributing to its $800 billion annual trade deficit with the rest of the world.
"It's going to be different," said the president-elect. "I mean Germany is a great country, great manufacturing country - you go down Fifth Avenue, everybody has a Mercedes-Benz in front of their building, right?
"The fact is that it's been very unfair to the US. It's not a two-way street. How many Chevrolets do you see in Germany? Maybe none - not too many. It's a one-way street. I want it to be fair, but it's got to be a two-way street."
European companies such as BMW considering opening plants in Mexico should be aware that they will face a 35 per cent tariff on exports to the US, he warned.
On the future of the European single currency, Mr Trump said: "Well, it's doing OK. I mean, you know ...
"But I do think keeping it together is not going to be as easy as a lot of people think. And I think this, if refugees keep pouring into different part of Europe, I think it's going to be very hard to keep it together because people are angry about it."
Asked whether the US would prefer a strong EU or stronger nation states, he replied: "Personally, I don't think it matters much for the United States. I never thought it mattered. Look, the EU was formed, partially, to beat the United States on trade, OK? So, I don't really care whether it's separate or together, to me it doesn't matter."