The Democratic proposals, coming two days before historic impeachment hearings, sought to highlight policy differences with the embattled president before a key bloc of voters.
"I think leadership plays a huge role so absolutely I'd seek to name a woman to lead VA," he said.
Trump praised the strength of the U.S. military and the death of Islamic State leader Abu Bakr al-Baghdadi, saying the nation's veterans "risked everything for us. Now it is our duty to serve and protect them every single day of our lives."
More than 100 protesters booed, some holding black balloons that read "support our troops, impeach." In a liberal city where Trump is deeply unpopular in spite of his roots there, a nearby building's soaring windows were adorned with signs reading "IMPEACH" and "CONVICT."
Veterans overall have strongly backed Trump throughout his presidency, though views vary widely by party, gender and age, according to AP VoteCast, a survey of 2018 midterm voters. In particular, younger veterans and women generally were more skeptical of Trump, who received multiple draft deferments to avoid going to Vietnam.
Former Vice President Joe Biden, whose late son Beau spent a year in Iraq with the Army, stressed that he would "restore trust" in VA.
Taking aim at Trump's stalled progress in reducing suicide among veterans, Biden pledged to hire more VA staff to cut down office wait times for vets at risk of suicide to zero as well as continuing the efforts of the Obama-Biden administration to stem homelessness.
About 20 veterans die by suicide each day, a rate basically unchanged during the Trump administration. Trump earlier this year directed a Cabinet-level task force to develop a broader roadmap for veterans' suicide prevention, due out next spring.
"Our veterans deserve leaders who will fight for them as ardently and as forcefully as they have fought for us," Biden wrote in a Veterans Day statement with his wife, Jill.
In a jab at Trump, Vermont Sen. Bernie Sanders released a video highlighting his role in working with the late Republican Sen. John McCain, a decorated war hero, to pass legislation that included the Veterans Choice program in 2014.
Trump routinely takes credit for being the first to enact the Choice program. What he actually got done was an expansion of the program achieved by McCain and Sanders, a former chairman of the Senate Veterans Affairs Committee.
Speaking at an event focused on seniors and veterans in Des Moines, Iowa, Sanders pledged to combat efforts to privatize the VA and assured a questioner that he would end the "very ugly practice" of deporting military veterans who are not U.S. citizens.
"How cruel is it that when people put their lives on the line to protect us," they are deported, he said.
As president, Sanders promised he would build upon his past legislative efforts by making it easier for veterans to get into the VA system.
He joins Buttigieg and Massachusetts Sen. Elizabeth Warren in urging increases in doctor pay to attract top VA candidates and fill 49,000 VA positions that have sat vacant as the Trump administration promoted private health care options.
During a Veterans Day speech in Rochester, New Hampshire, Buttigieg reflected on his own military path, while taking some digs at Trump.
"Having seen the outrage of Americans willing to put their lives on the line for this country having their careers threatened by a president who avoided his own chance to serve, yes, we are going to end the transgender military ban right away," Buttigieg said.
He added later in the speech that the VA needs to be depoliticized.
"We're going to have five-year terms for key positions so that decisions are made based on what is best for veterans and not based on whoever last spoke to the president during a golf game or made the right campaign contribution," Buttigieg said.
During a campaign stop in southeast Iowa, Biden noted that he carries with him every day the totals of those who have died as a consequence of wars in Iraq and Afghanistan.
As of Monday, he told about 200 people at a private college in Oskaloosa, the total had reached 6,900. "Every single one of those fallen angels leave a broken community behind," he said.
Tens of thousands more, however, have returned home with post-traumatic stress disorder, Biden said. "They are in trouble and they deserve every single thing we can give them," he said.
Yen reported from Washington. Associated Press writers Alexandra Jaffe in Des Moines, Iowa, Tom Beaumont in Oskaloosa, Iowa, and Zeke Miller in New York contributed to this report.