The word ashita is purely Japanese. The spelling 明日 comes from Chinese.
A note about reading types
For any word where the reading is the 訓【くん】読【よ】み, the word itself as pronounced is (almost always) from a native Japanese root. In these cases, it's important to recognize that the spelling and the underlying Japanese word are independent: the written form is from Chinese and was applied (usually) without regard for any phonetic similarity between the spoken Chinese form and the spoken Japanese form.
The Japanese word ashita is one such example. This reading ashita is a 訓【くん】読【よ】み from Japanese roots. 訓【くん】読【よ】み literally means "meaning reading", and comes from the Japanese perspective that "this Chinese spelling has this meaning in Japanese".
The 明日 spelling also has two 音【おん】読【よ】み, myōnichi and meijitsu, both ultimately from Middle Chinese //mˠiæŋ ȵiɪt̚// (though meijitsu appears to be a later borrowing, possibly influenced by a precursor to modern Min Nan pronunciation //miâ-ji̍t//). 音【おん】読【よ】み literally means "sound reading", and is from the Japanese perspective that "this Chinese spelling has this sound in Chinese".
The derivation of ashita
This word has been around for a very long time, appearing already in the Man'yōshū and Nihon Shoki of the early 700s, two of the oldest long-form Japanese texts. It stands in close relation to the words asu, asa, and asatte, and all are likely cognates.
- ashita (明日, more rarely 朝)
Originally referred to "morning", more specifically "dawn" when the night has ended. Contrasted with 夕べ【ゆうべ】 "evening, sunset". Over time, the sense shifted from "morning" in general, to "the next morning from now" → "tomorrow".
The phonetic development is lost to history, with no obviously clear roots. More on that below.
- asu (明日)
In the earliest texts, such as the Kojiki of 712, this word already referred specifically to "tomorrow".
- asa (朝)
Originally mostly synonymous with ashita, although asa never developed the same "tomorrow" sense. Asa was also used in many compounds and less often on its own, whereas ashita was a standalone noun.
- asatte (明後日)
Asatte with the geminate (double) //tt// is a comparatively recent term, only appearing in texts from the late 1600s. This evolved as a sound shift from earlier asate. That form appears in texts from the late 900s, still younger than the other terms in this list.
Derivationally, this appears to be asa + some suffix -te. Some sources suggest that this word arose as a contraction, possibly of asa + sarite, the conjunctive -te form of verb saru (去【さ】る, "to become that time").
User Nanigashi commented with a link to Gogen Allguide. While that site is good for some terms, its entries also include speculation that doesn't conform to known patterns of Japanese phonetic shifts, and their entry for ashita is one such problematic entry.
Gogen Allguide suggests that ashita somehow derives from ake (in reference to "dawn" as the "opening of the day") + shida ("time", a rare ancient Eastern Old Japanese word). However, there is no known and accepted mechanism whereby the -ke would simply vanish, nor is there any clear reason for the shida to become unvoiced shita. It is also a bit odd for a rare Eastern Old Japanese word to become the basis for a modern mainstream Japanese word; ancient Eastern dialect forms generally weren't adopted into the mainstream vocabulary.
One clear commonality in all of these terms is the initial as-, which appears to be a root form. Interestingly, adjective 浅【あさ】い, which usually means "shallow", also has a sense of "not much time has passed: still early". While the exact mechanics for deriving ashita from this hypothetical as- root remain unclear, I think this adjective is probably a cognate with these various "morning" terms.
If the above does not address your question, please comment and I can update.