Your only real option without replacing the windows would be to paint them. You can source paint that works for vinyl windows, however I'd recommend professional application for a variety of reason. That said, those windows look like an aluminum exterior as opposed to vinyl.

Secondly, is the exterior facing side of the grids between the glass actually green? That's extremely uncommon if so, its hard to tell in the picture. While using heavier screen could mask that, it will also cut a lot of natural light. You could look into adding some sort of applique to the exterior. There are companies that specialize in decorative glass appliques such as "leaded glass" look, etc, I'm sure they could come up with something to install mimicking your grid pattern and covering the green. You could actually source the material used for SDL grids by manufacturers (installed with two sided tape), but it will be a tedious install and there may not be clearance where the sashes overlap (although it looks like there is). Installing "flat bar" material wouldn't be all that bad to complete DIY. there are kits that can be purchased as well, something like this: https://www.amscousa.com/Window_Hardware/Grille/Window_Grille_Kit.htm 

With all of that said, to get these looking nice on the outside with a pro paint job and whetever is applied to the glass won't be an insignificant amount of money. I'd make sure that the windows are good enough that you would keep them for another decade+ before spending that.

Instead of bright white go with creamy white so as not to contrast as much with the beautiful dark green.