Previously, Mr Jones warned Wales was "unlikely to consent" to a deal unless certain conditions were secured.
Welsh Secretary Alun Cairns said it was "ultimately a decision" for Prime Minister Theresa May.
Mrs May has said she wanted the Welsh Government to be "involved and engaged" in the negotiations.
During a trip to Chicago last week, Mr Jones said talks would fail if it "becomes a two-way dialogue between Brussels and London".
He said all four of the UK's parliaments and assemblies should have to "agree to any deal the UK government comes to".
Mr Jones said he could not "envisage consent being given by Wales" unless access to the European Union's single market and protection for the rights of citizens from other EU countries were secured.
The call was interpreted by some as a threat to block a future Brexit deal.
But speaking to the BBC's Sunday Politics Wales programme, Mr Jones said he "never called for a veto".
He said: "What I have said is that I think it's right that all four parliaments should ratify, agree to any deal the UK government comes to.
"It wouldn't be right for the UK government to negotiate in terms of farming and fisheries when they have no role in farming and fisheries in Scotland, Wales or Northern Ireland.
"If the UK government want to get as much support as possible across the UK for any potential deal, they're going to have to seek support in Scotland, Wales and Northern Ireland as well."
Mr Cairns said it was ultimately a decision for the UK government's cabinet and the prime minister.
"The prime minister has tasked me to look after Wales' interests and in doing so I will engage with the first minister but the first minister will also want to engage with other administrations around the UK as well as with the UK government," he said.
"We will come to a view that serves the whole of the United Kingdom and it's my job to make sure that Wales is absolutely at the centre of those discussions."
Asked what he thought were Wales' priorities in the Brexit negotiations, Mr Cairns said: "It's about making sure that immigration controls are appropriate, that meets the needs of the community in Wales but also of Welsh business.
"It's about access to the markets and ensuring, of course, that we've got access to the new markets that are emerging.
"Australia, New Zealand, India, north America, Singapore - they've all indicated positive discussions. That will be so important to our future prosperity and trade opportunities."
First Minister Carwyn Jones has said he never called for Wales to veto the UK's future Brexit deal.