Well you could use chalk paint, but it will be easily scratched!
The peeps at my local Sherwin-Williams store told me if your primer is good you can. He recommended this primer. https://www.homedepot.com/p/Zinsser-1-Gal-Extreme-Adhesion-Interior-Exterior-Primer-2-Pack-348829/308297874?source=shoppingads&locale=en-US&mtc=Shopping-B-F_D24-G-D24-24_22_PRIMERS-Multi-NA-Feed-PLA-NA-NA-FY21_Multi_PLA&cm_mmc=Shopping-B-F_D24-G-D24-24_22_PRIMERS-Multi-NA-Feed-PLA-NA-NA-FY21_Multi_PLA-71700000074506538-58700006445219702-92700058237776977&gclid=Cj0KCQiApsiBBhCKARIsAN8o_4hZBP_CHg5-FdfaSpfFASM5OpuWaPqRHiPLevhwDF2VBmpucs-cbe8aAncJEALw_wcB&gclsrc=aw.ds
Yes, there are lots of DIYs that show it. I think it would wear best on a backsplash, where it doesn't get a lot of use...less chance to get scratched. https://www.google.com/search?ei=8wAzYIOrLeas5NoPqN60yAc&q=whitewash+travertine+tile+diy&oq=whitewash+travertine+tile+diy&gs_lcp=Cgdnd3Mtd2l6EAMyBQgAEM0CMgUIABDNAjIFCAAQzQIyBQgAEM0COgcIABCwAxBDOgcIABBHELADULMcWKAgYLIlaAFwAngBgAGYAogBoAWSAQUzLjEuMZgBAKABAaoBB2d3cy13aXrIAQrAAQE&sclient=gws-wiz&ved=0ahUKEwjD8JyKpPzuAhVmFlkFHSgvDXkQ4dUDCA0&uact=5#kpvalbx=_-QAzYKTELdmv5NoPwdyv8Aw29
Hi Doris: You can, but make sure that the surface isn't shiny, otherwise the paint won't stick. You can either prime it or sand it lightly. The new paint has to have a surface that it can stick to, and a shiny surface will not allow that.
You can paint it. Use oil based paint and prep it will with cleaning. I'm not sure if you can "white wash" but you can definitely paint it white.
here is some info Doris
Stone is very porous so i'd use a primer for stone
Prime it first and then you can
It won’t stick it will peel off
They can also be painted or whitewashed. They should be sealed if you want to protect the stone and the stain. To whitewash travertine tile, create a white paint and water ratio of 1:1 and paint it on each tile. ... When remixing, make sure that you keep the same ratio so that the whitewash is evenly distributed
use a primer for stone
Heres a Vid that should get you going Good Luck
Make sure to use a primer first.