“We are certainly not backing down from the challenge,” he said.
“We will always have a place for Northern Ireland in the United Kingdom, but it is not an option.”
Mr Robinson was sacked as Northern Ireland’s top civil servant in the wake of the Brexit vote and later quit the party.
He said he hoped Northern Ireland would be allowed to stay in the EU as it is, but the DUP will not be giving up on the union.
“I am sure that if we have to make the choice, it will be that Northern Ireland is a part of the United States of Europe,” he told BBC News NI.
“That is not something we will ever abandon.”
DUP leader Peter Robison said the party was still committed to the UK.
“If the Prime Minister, who is the leader of the Conservative Party, has the vision that we can take our place in Europe and have a very, very strong, strong, secure position in the European Union, then I think that we have a strong argument,” he added.
Mr Robinson, who has been appointed as Northern Irish ambassador to the United Nations, said he was not worried about the prospect of Mr Corbyn replacing him.
“As I said before, we have the opportunity to be part of an institution that is a beacon for people around the world,” he commented.
“What I am more worried about is that people look at what is happening in the UK and see that they are getting a better deal and we are not getting a real deal, and I am really concerned about that.”