This is a closeup on one of the leaves

I would spray with a fungicide immediately. Spray all the leaves well asap. I had a similar thing on some pumpkin leaves. It was white powdery patches, and when I sprayed it stopped it from spreading. I used Daconil.

Is there any homemade remedy I can use? Like baking soda?

Powdery mildew. Advanced case. Your vines will probably die fairly soon, but you should get your melons harvested before then.

No, there is not a reliable home remedy like baking soda. We have conceived and made fungicides and chemicals for a reason, because nature likes to win. Fungicides are not harmful to humans unless directly ingested (I'm assuming, since we have been using copper based sprays for the last 70 years and no one has died from it that we can say definitively).

both ltilton and I are from illinois, and for the last 3 years we have both watched as our precious melons have died for mysterious reasons.

This year I actually had two melons ripening, cantaloupe, and the racoons tried to cart them off. Dragged the melons a good ten feet, tearing the skin... so no muskmelon for me this year, again. (I shouldn't say that, I still have a chance, with one tiny melon still trying to make it. I need to guard it with trip mines and large caliber guns :)

When nature keeps slapping you in the face, you start to give up the idealistic thoughts of gardening 100% organically. If you want perfect melons, organically grown with no chemicals or fungicides, where nothing will hurt them and disease won't touch them, then you need to grow the melons in their native land: Persia.

Sorry if I sounded too... on the nose, best intentions, you know