While English spelling isn't very consistent, "meme" IS pronounced exactly as it's spelled. The rule that a single "e" after a single consonant lengthens the vowel before the consonant, so "meme" is pronounced "meem".
Other words also follow this rule, such as "make", "hate", "site", "bite", "dote", "fire", etc.
Of course, English being English, there are exceptions to this rule. "One" would logically be pronounced like "own", but it's actually pronounced like "wun". You'd expect "done" to rhyme with "dome", but it's actually pronounced "dun".
A single 'e' at the end of a word on its own is almost never pronounced in English, except in some loanwords e.g. "cafe". The only word I can think of where a single "e" makes its normal sound (as in "red") at the end of a word is in one of the two pronunciations of "the".