Ich hab mich in dich verliebt.
I know it means
I've fallen in love with you.
But since the definition of verliebt is '"falling in love", what is the use of mich in?
In other words, can I say
Ich hab verliebt dich.
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Verlieben as verb is reflexive, you must always use a reflexive pronoun like mich, dich, sich...
So in your example, the core message is
Ich habe mich verliebt.
In order to include the object of your affection, you need the matching particle, in this case in dich. Note that you can't safely translate particles from one language in another, you most likely go wrong. Also, the verb positions remain consistent, i.e. flexed verb in second, participle (usually) at the end, so one possible sequence is:
Ich habe mich in dich verliebt.
Other options, but with slight shift of emphasis are for example:
In dich habe ich mich verliebt.
Verliebt habe ich mich in dich.
In short: No, you can omit neither mich nor in.