Not sure, but it might be possible to install the .net framework on OS X. [That's just from a quick Google search](https://www.google.com/#safe=off&q=.net+mac), and I don't truly understand this stuff, but take a look. [I believe you can also use Wine to run it.](https://appdb.winehq.org/objectManager.php?sClass=application&iId=6551)

[Even though this isn't directed at your use case](https://apple.stackexchange.com/questions/162479/paint-net-alternative-for-mac-os-x/162483), take a look at this post about alternatives. [This post](http://graphicdesign.stackexchange.com/questions/6552/what-are-the-best-mac-applications-for-developing-pixel-art) also suggests alternatives, including [Pixen](http://www.pixenapp.com), [PikoPixel](http://twilightedge.com/mac/pikopixel/), [Piskel](http://www.piskelapp.com), [Aesprite](http://www.aseprite.org), and [setting up GIMP to work well with Pixel art.](https://www.youtube.com/watch?v=PONe4IIYSnQ) Forewarning, I don't do much pixel art (I've always just used GIMP) and therefore can't give you anything other than other's advice.