If Buyer "changed the closing date" after the parties agreed and signed the same version of the contract, that act has no legal significance. The date in the signed document is the contractual closing date. If Buyer does not comply with that aspect of the contract, he will have breached the contract, the sale falls through, and presumably forfeits his "earnest money" (assuming there is such a thing in the deal).
If Seller has simply made an initial offer which Buyer accepts (without a counteroffer), Seller has not yet agreed and there is no true mutual acceptance: Seller can reject the Buyer's proposal and offer some modified term. Once both parties agree and have signed the same version of the document, a contract has been formed. By changing the closing date, the terms have changed and Buyer has not signed (agreed to) the version that Seller signed. Buyer then has the same option to accept, reject or counteroffer, until one of them rejects or both of them accept the same thing.
There is nothing inappropriate about modifying terms of a real estate contract: it it quite ordinary and common.