Former Vice President Joe Biden says Tuesday night's win for Democratic candidate Doug Jones in the highly anticipated special election for Senate in Alabama "sends a message" to the people of the United States amid persistent partisan division and polarization. Biden praised Jones, a former prosecutor, for focusing his campaign on issues that really matter to the people of the state.

"It sends a message that not only do they reject all that [Roy Moore] stood for ... but also Doug talked about the issues he cares about," said Biden on "CBS This Morning."

Asked if the message sent by the vote in Jones' favor is one of unity, Biden replied, "I think it is."

He added, "Some of the happiest people in America today are the Republicans in the Senate. The divisiveness that was caused within the Republican Party by [Steve] Bannon and all this sort of bile that comes out of the administration, it's just — I know the majority of the members in the Senate on Republican side and are decent people, they're not comfortable in this territory."

Biden, who says he spoke to Jones three times on Election Day, said the advice he gave to the candidate was to "focus on the issues."

"If you noticed the bulk of what he did was that, and last night — instead of a victory lap like a lot of Democrats may have wanted to do and go out and say this is a big loss for the president, which it is — he talked about bringing people together," said Biden, who has known the senator-elect since Jones was a young volunteer on Biden's 1988 presidential campaign.

Biden, who writes in his new book "Promise Me, Dad: A Year of Hope, Hardship, and Purpose" about overcoming grief after the death of his son Beau, as well as about the current state of politics, says the ability to find the good in people of the opposite party is "absolutely, positively necessary."

He says amid the current political climate, people just "don't know one another."

"No matter what their position is on an issue, you don't start off disliking them, you actually have a connection," he added.

Meanwhile, Biden, a vocal critic of the Trump administration since leaving office, says the Trump White House is "on the cusp of doing good" on a number of things, namely how the United Nations and U.N. Ambassador Nikki Haley are handling the North Korean crisis.

"This is the time to engage and have more diplomacy," he suggested.

Biden says in light of Secretary of State Rex Tillerson's openness to potentially negotiate with the North Koreans without pre-conditions, the idea of containment of North Korea's missile program may be inherited by the next president.

But Biden took a jab at Mr. Trump's leadership style, saying he's "devoid of any depth of knowledge in foreign policy" and "not a student of detail."

"I wish the president would be president and stop being so consumed with himself and his image."

In particular, Biden called the president's tweets directed at Sen. Kirsten Gillibrand, D-New York, "disgusting." He added, "I never thought I'd say that about a president. It's disgusting, and our children are listening."

Biden suggested that Gillibrand is a name to keep an eye out for in the 2020 presidential race. Though he refused to rule out running himself, he also said he wants to let a new crop of "really talented people" in the Democratic Party to step up to the plate.

"We need somebody who's going to be really up to speed and totally confident in foreign policy and reach out across the aisle," Biden said, adding "you can't run this democracy without consensus."

Asked far as rumors about a run himself, Biden said, "If I got offered the job this moment by the Lord almighty, I'd say no because I don't know where things are going to be in the next two years with my family."

"I'm not saying I wouldn't run, but I've given my word there's nothing I'm doing now to put myself in the position to do that. But I'm going to be involved."