Biden's stance that he would codify Roe vs Wade has been misconstrued and Republicans claim that he and Democrats support late-term abortions.
Trump might have been referring a bill proposed last year by Democrats in the Virginia legislature, which was unsuccessful. The bill sought to make it easier for women to obtain abortions late in pregnancy if the mother's physical health or safety were at risk. The proposed legislation prompted Senator Ted Cruz, a Republican from Texas, to falsely claim that Democrats support abortion up until the moment of birth. Many other Republicans, including Vice President Mike Pence, have echoed the claim.
During the 2020 Presidential debates, Biden said that he would codify Roe vs Wade, the Supreme Court ruling that had made abortion legal nationwide, into federal law. This would limit abortions to the first 20 to 24 weeks of gestation. In their 2020 party platform, the Democrats have noted that they believe every woman should be able to access high-quality reproductive health care services, including safe and legal abortion. But that does not mean the Democrats, or Joe Biden, call for women to have an unfettered right to terminate pregnancies up until the point of birth.
Supreme Court's rulings in Roe and Planned Parenthood v. Casey say states may ban abortion after the fetus reaches viability, the point at which it can sustain life, which happens at or near the end of the second trimester. However, states with such bans must allow an exception to preserve the life or health of the mother. When the Washington Post asked whether Biden supported restrictions, a Biden campaign representative told them that Biden believes in the terms laid out by Roe and Casey. The Post also states that around 1% of abortions happen after the point of viability, so the cases at issue in this claim would be relatively rare.
Based on the evidence listed, we conclude that this claim is misleading. Joe Biden does not believe in or support abortions up to birth.