He said "nothing is going to stop" his administration moving forward and assuming power on 20 January 2021, despite President Donald Trump's refusal to concede the race for the White House.
Mr Biden said that his transition is "well under way" and that he is reviewing potential cabinet picks and other positions.
He referenced telephone calls with six world leaders, including Boris Johnson, saying the response had been "very fulsome and energetic".
"I'm confident we will be able to put America back in a place of respect it had before," he added.
Mr Trump has reportedly blocked his Democratic rival from receiving the intelligence briefings traditionally shared with incoming presidents.
But Mr Biden downplayed the impact of the Republican resistance, which he said "does not change the dynamic at all in what we're able to do".
In the major speech Mr Biden set out his ambitions for US healthcare, saying the Affordable Care Act is a "matter of life and death", and reiterating his commitment to protecting the legislation.
Mr Biden was joined by vice president-elect Kamala Harris for the talk in Wilmington, Delaware, on the Trump administration's lawsuit to overturn the act, and his plan to expand access to quality, affordable healthcare.
She said Mr Biden "won the election decisively", and that "every vote for Joe Biden was a statement that health care in America should be a right, not a privilege".
Mr Biden said the Trump Administration's efforts to repeal the 10-year-old law, popularly known as Obamacare, would "rip" healthcare away "in the middle of the worst pandemic in a century", leaving millions of people who have pre-existing conditions without coverage.
Nevertheless he promised a "dramatic expansion" of healthcare provision and said his team is "fleshing out the details" on a plan to provide universal and cheaper care "as soon as humanly possible".
"I will protect your healthcare like I protect my own kids, my own family," he said.
Earlier, the Supreme Court indicated it was likely to leave in place the bulk of the act, including key protections for pre-existing health conditions and subsidised insurance premiums that affect tens of millions of Americans.
Chief Justice John Roberts and Justice Brett Kavanaugh appeared in two hours of arguments to be unwilling to strike down the entire law - a long-held Republican goal that has repeatedly failed in Congress and the courts.
A week after the 2020 election, the justices heard arguments by telephone in the midst of the coronavirus pandemic in the court's third major case over the act.
Republican attorneys general in 18 states and the administration want the whole law to be struck down.
The case represents the latest Republican legal attack on the 2010 law, which was the signature domestic policy achievement of Democratic former president Barack Obama, under whom Mr Biden served as vice president.
The Supreme Court fended off legal challenges to it in 2012 and 2015.
It has a 6-3 conservative majority after the Republican-led Senate last month confirmed Mr Trump's third appointee, Amy Coney Barrett.
Analysis: A damning indictment
By Sally Lockwood, news correspondent, in Delaware
We've become used to Donald Trump's lack of convention but a week on from the election it's still hard to believe America's president is refusing to accept the result.
For the first time we've now had an insight into his successor's response... "It's an embarrassment".
A pretty damning indictment of Donald Trump's undignified reaction to losing the election.
When questioned over this lack of co-operation in the transition of power Secretary of State Mike Pompeo responded: "There will be a smooth transition to a second Trump administration". It wasn't clear if this was a joke or if Donald Trump and his team are deluded enough to think they can somehow remain in the White House for another four years.
Yet Donald Trump still shows no signs of conceding and with that comes significant challenges for President-Elect Joe Biden. By not triggering the official transition process, he is unable to access millions of dollars in federal funding and classified intelligence reports.
You would expect a new president to be across this sensitive information before reaching the Oval Office and national security is at stake. Biden's response to this lack of resource - it would be nice to have it but we can manage without it.
You would also expect the Biden transition team to be working from a federal building in Washington - instead, he's still operating out of a small theatre in his hometown of Wilmington, Delaware.
The optics are nothing short of extraordinary.
Nevertheless, while the White House appears to be in denial and disarray, Joe Biden's tone was calm and confident.
The transition of power is pressing ahead with or without Donald Trump's consensus.