It sure is fun watching everyone talk about and show photos of their melons- especially new growers. In my experience, it isn't uncommon for melons to stop growing and then start back with no explanation for why they do either. Usually, though, mine will show very obvious increases in growth about a week after application of fertilize (plain 15-15-15 works wonders for me). One of the things that makes a huge difference to me and that I love to do is to try and apply a good dose of fertilize right before a good rain. About 3-4 days later you'll see bright green tips on all the vines and the melons will grow too. I've heard a lot debate about whether excessive water and/or fertilize affects sweetness or not. I won't get into that debate but I will say that for me giving a fair amount of both (fertilize more than water, and I don't water mine unless it's desperately dry) makes such a huge difference in production of vines and melons that I can't resist doing it and usually have really good, sweet watermelons. I highly respect fruitnut and he's a big believer in shorting water to increase sweetness in all fruits. Since I only water in extreme droughts, and my plants are all outside, I just can't really control water amounts and just get what comes. But in general, my watermelons do better in hot, fairly dry summers that just have occasional rains.
Don't worry about stalling growth. It is true that sometimes a cessation of growth is the first step in a watermelon spontaneously aborting (this can happen almost any time, but gets less common the larger a melon gets). But there is not a thing you can do about it so I wouldn't worry. Its just as common that they will take off again or that they have really just slowed down but not truly stalled. The bigger a watermelon gets the slower it grows, so it can be a bit deceiving.
BTW: I wouldn't thin watermelons on a vine! Yes, if you are very seriously about growing only giant watermelons, then you can pluck all but 1 or two melons off a vine to get larger ones. But listen to me on this because I've tried it several times....the difference in size you will get by thinning off some or most watermelons is just not nearly as much as you think. Plants will do this naturally so that it will usually end up with about the optimal amount of melons it can support. If too many get pollinated some will be aborted. But if you pluck off good, viable melons yourself you will most certainly be reducing the net poundage of melons at harvest. In other words, even an over burdened melon plant will produce a total poundage of fruit that will be considerably higher than only 1-3 larger melons would add up to. In fact, my experience is that the size of melons on a vine with 4 melons isn't nearly as much higher as you would think compared to if you thin it down to 1-2. Even champion, record chasing growers will tell you there is much, much, much more to getting giant melons than just the number of the vine.
These are just my own experiences....others may have other ones. Either way, nice to see so many people having fun growing watermelons. I'm about to pick my first cantaloupe of the year which is actually about 1-2 weeks early for me.