For years, people have speculated about Amazon rolling out a health care service as part of Amazon Prime. Notable early Amazon investor John Doerr is already calling it "Prime Health."

So far, that hasn't happened. But the former head of Amazon Prime, Rob Schwietzer, who was in charge of scaling the product to millions of people, is now working in health care. He left Amazon in 2013, and for the past year, he's been leading product at 98point6, a Seattle-based start-up that is looking to build the next consumer health brand.

"Health care is really hard," Schwietzer told CNBC. "But I do believe that people shouldn't have to make decisions about whether they should go see a doctor when they need one, or go buy groceries for their family."

Schwietzer has teamed up with CEO Robbie Cape, a Microsoft veteran who created Cozi, a family planning software venture, then sold it to Time. Back in 2011, Cape contacted Schweitzer out of the blue to get some advice on subscription businesses, as Prime was one of the few services millions of people were willing to pay for every month.

To his surprise, Schwietzer responded and the pair had lunch.

Years later, when Cape had the idea for 98point6, he reached out to his old friend Schwietzer. He had his misgivings about health care after a stint in consulting early in his career, but Schwietzer said he saw a big opportunity to bring "price, quality, and convenience" to the sector, similarly to what Amazon did for retail.

"Why I think Amazon does so well," Schwietzer explained, "is that where most companies will look at something and say 'we'll give it to you fast or inexpensive,' Amazon has always said 'why not both?'"

Meaningful innovation in health care could not only create a huge business -- health care spending has now reached $3.5 trillion -- but also make a meaningful difference. Studies have shown that improving access to high-quality family doctors can help drive down health care costs and improve patient's health outcomes, but not everyone can afford it or reliably access it.

That's because there's a shortage of primary care physicians and nurses in the U.S., which is only getting more pronounced with the aging population. People in rural areas are often hours away from the nearest clinic, and many will forgo care until they end up in the emergency room.