Is there a Way to Make an Image of a Sphere into an Equirectangular (Mercator) Projection?

A search for "Map Projection Software" will give you lots of options, like this one from NASA.

Just ignore all the hits relating to "Projection Mapping Software". That's completely different.​

Are you assuming that the first image is a full sphere including the hidden detail around the back?

I guess you'd start by looking at Mercator Projection

Mercator projection - Wikipedia

And search for software that would produce it.  Being such a fundamental part of map and chart making, I'm thinking that there will be tools available, but I'm wondering what format the sphere would need to be in?

[EDIT]  I have placed the images into your initial post, and added the word Mercator to the subject line.  FYI  It is easy to add images to this forum's posts.  Just paste from the clipboard with Ctrl v (Cmd v).  1000 pixels on the long side is plenty big enough.