Instead of saying it's nice to meet you, should I say pleased to be of your acquaintance? ?

You can say that if you wish.  It’s an alternative, quite formal greeting.  But people do like to hear a different turn of phrase in English.  Just taking pleasure in the greeting and language itself.  It sets a pleasant light hearted tone for continued conversation.

Only if you want people to think you're mad.

It's excessively formal.

Post in "words and wordplay " or "Etiquette"  not here please.

It is different in different countries. You are obviously in the US or Canada, the only countries where they say that. In England they say 'How do you do' to which the answer is 'How do you do'. It is just a greeting and no answer other than that is expected. 

I suppose that is because the English do not know on first meeting whether it is nice to meet you or not. 

Next time somebody says 'Nice to meet you' you can answer with 'The pleasure is all yours'. I bet he or she will not even notice. 

“Nice to meet you, to meet you nice“ always goes down well with a certain demographic.

Nice to meet you is fine Jerry, don't force it

either will work ............................