Speaking exclusively to the BBC as the polls were closing, Carney said he would only visit Washington when there was a "serious discussion to be had" that respected Canada's sovereignty.
Carney and Trump have since spoken and agreed to meet in the near future, according to the Canadian prime minister's office.
"The leaders agreed on the importance of Canada and the US working together – as independent, sovereign nations – for their mutual betterment," the statement reads.
Trump was also said to congratulate Carney on the election.
Since Trump's re-election to the White House, the US president has repeatedly mentioned making Canada the "51st state" of America, and earlier on Tuesday this was reiterated by the White House on Tuesday.
"The election does not affect President Trump's plan to make Canada America's cherished 51st state," White House deputy spokesperson Anna Kelly said.
Carney, who secured a historic victory for his Liberal Party in a snap election on Monday, said such a scenario was "never, ever going to happen".
"Frankly, I don't think it's ever going to happen with respect to any other [country]... whether it's Panama or Greenland or elsewhere," he added.
However, he said there was a "win-win possibility" for his country if it could secure a deal with the US and also build on trading relationships with the European Union and the UK.
Strained US relations
The US is a big market for Canadian businesses with roughly 75% of Canada's exports heading south.
Canada accounts for a much smaller 17% of US exports.
Canada is also America's largest foreign supplier of crude oil. America's trade deficit with Canada - expected to be $45bn in 2024 - was mostly driven by US energy demands.
Canada and US relations have been strained in recent months, driven by Trump's talk of a "51st state" and referring to previous Prime Minister Justin Trudeau as "governor" - the title for leaders of individual American states.
The US president has also sparked a global trade war in which Canada was one of the first countries targeted with tariffs.
Trump has partially imposed a blanket 25% tariff on various Canadian goods, along with 25% import taxes on all aluminium and steel imports, but has exempted products covered by a US, Canada and Mexico trade deal known as USMCA.
Canada has retaliated with some C$60bn ($42bn; £32bn) worth of tariffs on US goods.
Carney said talks with Trump would be "on our terms, not on their terms".
"There is a partnership to be had, an economic and security partnership," he said.
"It's going to be a very different one than we've had in the past."