No, you don't HAVE to grout. But, what you'll be creating are lots of tight crevices that dirt and liquid foods can get into, but you can't clean that dirt or food out of. Eventually, you'll end up with the tiles surrounded by dirt and rotting food. Why would you want to do that? In a kitchen, good hygene trumps appearance.
If you want to do away with the grout lines, then after setting your tiles, paint over the tight joints with a water based acrylic "varnish" like Minwax's "Polycrylic". It will be a milky white liquid that dries clear. I'm thinking if you can use the capillary pressure in the tight joints between the tiles to draw in a liquid plastic that will harden up inside that crevice, then you have the benefit of having a filled joint without seeing white lines between the tiles.
If it were me, I would dip an artist's paint brush in the Polycrylic and touch it to the joint so that capillary pressure pulls the plastic slurry into the joint. Hold a bright light close to the tiling when you're doing this. Once you see a sliver of light between the tiles (indicating the reflection off a liquid surface between the tiles) then the joint is full of liquid. Also, acrylics shrink substantially as they dry, so you may need to apply the Polycrylic more than once to fill in that shrinkage. In fact, since the stuff dries clear, I'd keep applying it until the reflection of light off the joint even when it's dry indicates that the crevice has been completely filled with plastic.
If you can't find Polycrylic, then any acrylic grout sealer will do the same job for you. TileLab makes a good one that I like a lot called "Gloss Sealer & Finish". It dries to a very hard very transparent film.
Good eyesight will be of some benefit here.