At his weekly press conference on Capitol Hill the Ohio Republican reiterated his belief that the problems that plague the VA go beyond one person.
"I admit I'm getting a little closer to calling on him to resign," Boehner said. But, he added, "this isn't about one person, it's about the entire system."
Boehner said he thought it would be counterproductive to insist Shinseki leave his post, only to then have to wait for the confirmation process to install a replacement.
"The general could leave and we'd have to wait months while the disaster continues," he said. "I don't want people to get confused about what the shiny ball is."
So far, the VA has dispatched its inspector general to conduct a full review into allegations that staff at VA health centers kept secret wait lists to disguise how long veterans were forced to wait to receive care. The scandal erupted last month after a whistleblower came forward to reveal more than 40 veterans had died while waiting for care in the Phoenix, Ariz., area.
"We've not just let [veterans] down, we've let them die," Boehner said, his voice rising. "This is awful stuff, and somebody ought to be held accountable for it."
Minority Leader Nancy Pelosi, D-Calif., at her press conference later Thursday, agreed that forcing Shinseki out would do little to solve the VA's problems, but neither would reactive efforts to fix them.
Pelosi said there are 2 million more veterans that have entered the VA system in the past five years alone, and who will, over their lifetimes, cost between $4 and $5 trillion to care for. She suggested that a major overhaul of the care system is necessary, perhaps even finding ways to get veterans care outside the VHA.
"Democrats share the president's outrage," she said. "But we must think in a better way."