How much time passes between each novel in "The Expanse" series?

Here's my approximate timeline in the format you requested, starting on Year 0 at the beginning of the first novel:

• Leviathan Wakes: 0 years – 0 years, 4 months
• Caliban's War: 1 year, 6 months – 1 years, 10 months
• Abaddon's Gate: 2 years, 10 months – 3 years, 4 months
• Cibola Burn: 5 years, 10 months – 7 years, 4 months
• Nemesis Games: 8 years, 10 months – 9 years, 4 months
• Babylon's Ashes: 10 years, 4 months – 10 years, 10 months

This was put together from references to time in the novels like the ones below. To keep the timeline a little simpler I mostly kept to a precision of months (as you say, there are no definite dates in the books and all dating is done relative to previous events, so the precision is low to begin with) and assume it fits together with a little rounding here and there.

A note on prologues and epilogues: These often take place some time before and after the bulk of the novel (for example, the prologue in Nemesis Games takes place one year before the first chapter). I have disregarded these in the timeline and don't consider them part of the "duration" of each novel.

Leviathan Wakes

The first novel starts at Year 0, as per your requests. The plot takes place over the course of a few months:

Earth's blockade of Mars was entering its third month.
_{Leviathan Wakes, Chapter Forty-Two}

Let's say four months total, as a guess.

Caliban's War

This book starts not more than 1½ years after the beginning of the previous book:

Eighteen months before, there hadn't been sides. The inner planets had been one big, happy, slightly dysfunctional family. Then Eros, and now the two superpowers were dividing up the solar system between them, and the one moon neither side was willing to give up was Ganymede, breadbasket of the Jovian system.
_{Caliban's War, Chapter One}

So let's say it starts 1½ years after Year 0. The book then spans at least four months:

Four months ago, hours before the first attack on Ganymede, a little girl with a life-threatening genetic illness was abducted from her day care.
_{Caliban's War, Chapter Thirty-Three}

Abaddon's Gate

The third book starts about one year after the conclusion of the last one concluded:

You've been doing your random appearing act for almost a year now, and you've never said even one thing that makes sense.
_{Abaddon's Gate, Chapter One}

The trip out to the Ring takes four months:

"Dammit," he said. "Should have confined them to quarters."
"For four months?"
_{Abaddon's Gate, Chapter Ten}

Then there's some more stuff that happens, the trip is delayed a little, etc. Let's say the book lasts half a year.

Cibola Burn

The fourth book initially begins around two years after the previous one ended:

Miller's ghost was an artifact of the alien technology that had created the gates and a dead man. It had been following Holden around for the two years since they'd deactivated the Ring Station.
_{Cibola Burn, Chapter Four}

Actually, more like 2½ years. First the colony was alone there for one year:

"So they've been living on Ilus for a year, and suddenly RCE shows up and tells them that, oops, it's really their planet?"
_{Cibola Burn, Chapter Four}

And multiple references are made to the fact that the journey from Medina to Ilus takes 1½ years:

"Old news. They been burning eighteen months. That's why we're out here."
_{Cibola Burn, Chapter One}

So, the Ilus colonists left our Solar system about 2½ years ago, which must have been after the previous book ended and the gates opened.

Now, here's a little quality check for our timeline. Holden says five years have passed since he first met Fred way back in Leviathan Wakes:

The five years since Holden had met him hadn't treated the man kindly.
_{Cibola Burn, Chapter Four}

My timeline says it's been a little over that (between 5 years, 6 months and 5 years, 10 months). We could chalk that down to Holden's faulty memory and approximation, but another piece of information also puts this book five years after Leviathan Wakes:

Dimitri Havelock had worked security contracts for eight different companies over thirteen years.
_{Cibola Burn, Chapter Three}

While in Leviathan Wakes, Havelock says this:

"That's crap. I was eight years in the orbitals and on Mars before I ever shipped out here. I worked on Earth maybe six months total."
_{Leviathan Wakes, Chapter Four}

I'm not sure if Havelock includes those six months on Earth as a part of those 13 years he's been working in security, but either way it's probably an approximation. So let's say that both Holden and Havelock round a little bit down, while I have rounded a little bit up. We're in the same ballpark!

Then there's a time skip in the book as the Rocinante travels for 1½ years to get to Ilus. The duration of the rest of the book after the Roci arrives isn't certain, but it can't be very long, as you'll see right below.

Nemesis Games

The very first sentence in the first chapter establishes the chronology here, after the Callisto attacks that take place in the prologue:

A year after the Callisto attacks, almost three years after he and his crew had headed out for Ilus, and about six days after they'd gotten back, James Holden floated next to his ship and watched a demolition mech cut her apart.
_{Nemesis Games, Chapter One}

So that decides both the starting point of this book and that the duration of the previous one (after the time skip) must be negligible.

The repairs on the Rocinante are slated to last for half a year:

"Six months is our current projection," Fred said, then sat in his chair with an old man's grunt.
_{Nemesis Games, Chapter One}

That repair is interrupted by some stuff happening, and then the final repairs are rushed a little, but there's no reason to believe it was rushed by too much to matter. Let's say the entire book takes place over the span of six months. It's probably a little more, but we'll round up the time between this book and the next anyway:

Babylon's Ashes

This book is harder to pinpoint. It takes place not more than a year after the large event in the previous book:

Thousands of people were dying every week from conditions that even a year ago would have been easily treated or cured.
_{Babylon's Ashes, Chapter Thirty-Eight}

Probably less, as the prologue is three months after the event, but it's hard to say how much time passes between the prologue and Chapter One, so let's just say one year after the previous book ended. There's a time skip of half a year late in the book though:

It had been six months now since the remarkable death of Marco Inaros and the great remnant of the Free Navy.
_{Babylon's Ashes, Chapter Fifty-Two}