How can I make a grey background white?

Try using the Curves edit mode. Click on the far right eyedropper and then on the greyest portion of the image.

You can then adjust the contrast or other options to increase the contrast with the other parts of the drawing.

You could try three things in the Edit > Adjustments panel

• Increase the Highlights slider: Then the grey areas will become overexposed and whiter, like this:

Additionally, shift the Exposure slider to make the image brighter.

• Or use the "Selective Color adjustment: Click the color picker in the Selective Color adjustment brick and sample a gray color in the background. Then increase the range and the luminance.
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On my display it is looking much better than on the screenshots I posted.

No. You would need a 3rd party editor for that kind of work. There are a lot of these. Search the App Store.