You can use the Majority Filter to remove cells that are surrounded by your green cells. This would also influence the boundary of your dominant green area as this tool works a local level.
In your sample image which I guess you have spoofed up you have dropped a wide range of hole sizes, how were you deciding which would stay and which would go? I can even see an example of green surround by black surrounded by green...
You give no indication of the size of the is problem, is it the whole of Brazil or an area in a small Icelandic national park? You should always state that sort of thing in your question as it will dictate any answer.
Having said that I can think of one approach, I'm sure there are others. Convert you binary raster to a polygon dataset, go into edit mode, select the polygon with the holes and then delete out the interior rings via the sketch properties window.