There is function in .js library called getBounds() http://leafletjs.com/reference.html
Is there something similar in the leaflet library with the R binding?
Yes, when used with Shiny. Look for "bounds" on this page: http://rstudio.github.io/leaflet/shiny.html
Most helpful comment
Yes, when used with Shiny. Look for "bounds" on this page: http://rstudio.github.io/leaflet/shiny.html