Are these images that you have done any adjustments on the Mac? If you take one of these images to the Windows PC to print it, does it look right on the monitor for that computer?
You should have the same printer controls on the Mac as you do on the Windows computer. Are you allowing Lightroom to control the color, or are you specifying that the printer manages the color? Or, are both managing the color?
All adjustments to arrive at the final image (to be printed) are on the Mac with LR.
With LR I have tried both 'Managed by Printer' and 'Adobe RGB' (in the 'Colour Management' section. If I select 'Others' (Instead of the default Managed by Printer) I get a long list of options : Adobe RGB, Display, and a long list of various Nikon & Epson alternatives. So far I have only tried the Adobe RGB and it was actually worse).
In PSE I leave it as Managed by Printer. But get the very different menu options in Mac vs Win (strange!).
The monitor on the Win PC is low quality but the image looks ok (tone, colour). And the printed image looks perfect - exactly as it was finished on the Mac.
You saved the image looks okay on the Windows PC (tone, color). But what about brightness? I think your monitor on the Mac is set too bright. You probably need to adjust at least the brightness, but a complete monitor calibration is probably in order. When you were using a Windows PC, did your prints from Lightroom turn out the way you expected them to?