Let's say that you are asking someone: "Let's meet at 5pm at place X?" and the person replies: lets meet at 5pm but at Y. Then if you are stuck on an emergency and cannot reply until 5pm. If you contact the person again at 5pm and they say "we agreed to meet at 5pm, why did you change plans?" Where you bind to that oral contract if you didn't agree to Y?
The four key elements of a contract, which are all required to make it valid, are: agreement, capacity, consideration, and intention.
Agreement is achieved by offer and acceptance. And that is where it already falls flat. Even if you count "Lets meet at 5pm at place X" is an offer, the reply "Lets meet, but not at X but at Y" is not an acceptance, its a counter-offer. If both parties just offered to meet and there was no other communication, then there has been no agreement at all. So you are already in trouble on the first point.
I am sure both you and the other person were capable of entering into a contract. If you are 18 years or older and you both understood the terms of the contract, that is.
Parties must exchange some value for a contract to be binding. That is why you see sale contracts for a symbolic € (or $). And that is the second point where this situation fails to meet the standard of a contract. Nothing of value was ever exchanged, thus there is no contract.
Not all agreements between parties are contracts. It must be clear that the parties intended to enter into a legally binding contract. [...] In social situations, there is generally no intention for agreements to become legally binding contracts (eg friends deciding to meet at a specific time would not constitute a valid contract).
Thus, the agreement to meet fails this test as well.
In short, no, there is no valid contract. The promise to meet fails to meet three of the four properties a contract needs to have.
I will set aside the problem that as posed, there is no exchange of valuable consideration so there's no contract. I assume the core question is whether silence (and no behavior that resembles acceptance) in the face of a counteroffer constitutes acceptance. The linguistic form is, b.t.w., not a question, it is an exhortation (a sub-type of of declaration).
You might call a tree surgeon in to size up a job and suggest that they come on Friday. The tree guy counters with Saturday, and you don't say anything, but you mentally reject that option. In the meantime you have another tree guy cut the tree down on Friday. The Saturday guy shows up, and suffers a loss because the tree is gone and he had to spend money on equipment and a crew. He reasonably relied on your apparent "acceptance" of the counter-offer.
In highly formal contracts with pages of clauses and signatures, the question of whether there is acceptance is reducible to the signature. Casual oral contracts are judged on the basis of "ordinary behavior".
Where you bind to that oral contract if you didn't agree to Y?
No, provided that the difference between places X and Y is material.
The interlocutor's choice of place Y constitutes a counter-offer. As such, it (1) constitutes a rejection of your original offer, and (2) requires your acceptance for there to be a cognizable agreement. The interlocutor is unable to establish (2). In fact, the interlocutor's reproach simply in terms of "we agreed to meet at 5pm" reflects his awareness that there was no agreement as to place.
You would be likewise precluded from alleging that you and the interlocutor agreed to meet at place X. That is because either your unforeseen emergency prevented you from ascertaining the interlocutor's response (be it an acceptance, plain rejection, or counter-offer), if any, or having read his counter-offer made you aware of the interlocutor's rejection of your offer/proposal.