Translation:I will quit the company tomorrow.
What you are suggesting is that grammatical tense is not necessary when a temporal description is provided. In the same line of thinking, does that mean "I did all the talking at yersterday's meeting" and "I do all the talking at yersterday's meeting" feel the same to you? To me, "I leave tomorrow" sounds like someone not fluent in English would say.
Not the best example, but "I quit the company tomorrow" sounds like it's past tense without any other context though is what he's saying. If it was a response to a question like "When will you quit?" you could drop the "will" because from the response because it was established already in the question.
Not downplaying your question, but it's purely an English grammar question, so I'll try to answer with that in mind.
Funnily enough, some linguists say English doesn't have a future tense, because 'will' is considered a modal auxiliary verb. It's also kind of funny in a way, because somewhat in the same sense as Japanese, English doesn't use a simple present typically. Basically we use only past and non-past with the exception of aspect (think -ing verbs aka continuous /progressive tense).
For instance consider 'to play'.
"I play chess" is passive, it always is true. "I play chess at 3 P.M." is future.
The only pure present form is "I am playing chess" which when you think a bit, is just a combination of past and non-past. The exceptions are the copula (is/are/am) when not using a verb e.g. "I am happy" and maybe some action verbs, currently "want" is the only exception I can think of, ("I want water" is completely present).