It said the shopping giant, which has faced allegations of poor working conditions, often tries to suppress union organising at its warehouses.
No UK Amazon warehouses are unionised, but by law workers could set one up.
Amazon said it respected its employees' right to "join, form or not to join a labour union" of their choice.
Unite's call comes after workers in Alabama in the US, voted against forming that country's first unionised Amazon warehouse.
Amazon - which would have had to negotiate on work rules and pay had it lost - said the union did not represent the views of most staff.
However, the RWDSU union, which organised the Alabama effort, accused Amazon of illegally interfering in the vote and lying about the implications of unionisation in mandatory staff meetings.
Amazon denies the claims but did hire anti-trade union consultants before the ballot.
'Without fear and obstruction'
Unite urged the shopping giant to sign a "neutrality declaration", guaranteeing UK and Irish workers it would not try to stop union organising.
It noted that in September 2020, Amazon had posted two job adverts for intelligence analysts to track labour "organizing threats" in the US. The ads were later withdrawn.
It also flagged Spanish media reports which claimed Amazon had used private detectives to spy on a strike at a warehouse near Barcelona on Black Friday in 2019. At the time, the shopping firm called the claims "irresponsible and incorrect".
In a letter to Amazon boss Jeff Bezos, Unite executive officer Sharon Graham wrote: "Although we do have members in Amazon, workers in your company are not currently free to join a union without fear and without obstruction and propaganda being deployed against them.
"So I am asking you to sign up to and abide by the declaration attached, which guarantees British and Irish workers the freedom to talk with and join unions without fear of retribution."
Amazon had its most lucrative year ever in 2020, helped by a surge in online shopping during the pandemic. But it also faced allegations over poor working conditions, as well strikes at warehouses in the US, Italy and Germany.
The shopping giant say it offers workers competitive salaries and benefits, and created 10,000 permanent jobs in the UK last year, taking its workforce to 40,000.
Amazon told the BBC its respected the right of its staff to be in a union.
A spokesman added: "Across Amazon, including in our fulfilment centres, we place enormous value on having daily conversations with each associate and work to make sure direct engagement with our employees is a strong part of our work culture.
"The fact is, we already offer excellent pay, excellent benefits and excellent opportunities for career growth, all while working in a safe, modern work environment. The unions know this."