Do natives use/say this phrase?
"The weather is melancholic today"
Yes, gloomy does come to mind. "Melancholic" refers to an emotional state, which people may have, but not weather. Now, "gloomy" is also an emotional state, but for some reason we can speak of "gloomy" weather or a "gloomy" night. We also wouldn't speak of weather as "sad." But, come to think of it, these days we often do. The weather this weekend is "pretty sad."
Perhaps others have ideas concerning which words can be used both for emotional states and for nature.