Goodness, forgot to update on this one. I've been away, in physical sense too.
So, waiting a week more did work out. The melon got ripe and tasty, less sweet than store-bought but with much stronger aroma.
Unfortunately, the weather started getting colder again and not many more managed to ripen. So, the experiment proves that it is perfectly possible to grow melons here. But it is very dependent on the weather during the summer.
The big mistake was the seed planting time, I did it too late, around late June as I've been waiting for properly hot weather. Should have done it a month earlier, then many more fruits would have the time to get good.