In response to Trump's announcement of the ban on Twitter on Wednesday, Shulkin said in an interview with 77WABC Radio, "Once a person signs up to defend our country, they have to know that the VA is there and committed to them through the end of their life."
"So that's not going to change; our commitment is there forever," he said, adding that the commitment will continue to apply to transgender veterans.
"I don't see this as a major complication for us," Shulkin said of the possibility of handling an influx of thousands of transgender troops who could be forced out of the military.
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"I think that it's important that veterans know that the VA is a safe place to get their care, that we value and respect the dignity of all our veterans, and we are not going to change our commitment or our mission to any veteran who chooses to get their services in the VA," he said.
"Any veteran that is discharged from service -- we will honor our responsibility to caring for them, so if there is an influx of veterans, that's something that we're used to seeing," said Shulkin, who acknowledged being blindsided by Trump's announcement.
In a series of Tweets on Wednesday, Trump said that the U.S. "will not accept or allow transgender individuals to serve in any capacity in the U.S. military." The president said he was acting after consulting with "my generals and military experts," although it was not immediately clear who they were.
The announcement appeared to come as a surprise to the Defense Department, which had been working up a long-term plan on the recruitment and retention of transgender individuals.
Shulkin said he also did not receive a head's up from the White House.
"I was not aware of the decision that was going to be announced," said Shulkin, the only holdover from the administration of former President Barack Obama in Trump's Cabinet.
Defense Secretary Jim Mattis, who earlier this month ordered a six-month review on transgender recruitment and retention, reportedly was on vacation when Trump began his early morning Tweets on the issue.
Pentagon reporters seeking reaction were told to call the White House. Navy Capt. Jeff Davis, a Pentagon spokesman, later issued a statement:
"We will continue to work closely with the White House to address the new guidance provided by the commander-in-chief on transgender individuals serving in the military. We will provide revised guidance to the department in the near future."
Acting Army Secretary Robert Speer also appeared to be out of the loop on Trump's transgender announcement. At an Association of the U.S. Army event, Speer said he was clueless on where the White House was "going with the policy" on transgender troops.
"I don't have my cell phone, so I'm not checking Twitter," he said with a chuckle. "To be tongue and cheek, Twitter is not a policy.
"I do know that we were encouraging the current SecDef to review the accessions policy on transgender, which he was doing, and that review and analysis was not done," Speer said.
"And that's the latest guidance, so I'm not going to get in between where the president or the SecDef is right now on where they are going on a future policy," he said.
In a statement Wednesday, Rep. Tom Rice, R-South Carolina, echoed the previous complaints of a number of Republican lawmakers that the military should not pay for the sexual reassignment surgeries of transgender troops.
"Military guidelines on the sex of its personnel, like all of its guidelines, should be designed to produce the most effective fighting force to defend our country -- period," he said.
"I leave it to the president and people in charge of the military how best to define that goal. However, I strongly oppose the obligation of the government to fund procedures for sexual reassignment or any other elected surgery," Rice said.
Defense Department officials have said that there are as many as 250 service members in the process of transitioning to their preferred genders or who have been approved to formally change gender within the Pentagon's personnel system.
The advocacy group Human Rights Campaign has estimated that there are about 15,500 transgender troops currently serving.
A Rand Corp. study commissioned by the Pentagon estimated that there are between 2,500 and 7,000 transgender service members on active duty, and an additional 1,500 to 4,000 in the Reserves and National Guard.
-- Matthew Cox contributed to this report.
-- Richard Sisk can be reached at Richard.Sisk@Military.com.