It said such a move would "reduce the burden" on passengers who could instead go to claims companies which would take a cut of any payouts.
Hundreds of flights were grounded by last Saturday's IT problems.
BA said that it had "no desire to be obstructive" and would fully honour its compensation obligations.
The problems, which continued throughout last weekend, disrupted the travel plans of tens of thousands of passengers around the world.
Under EU law, passengers can claim compensation from BA for travel disruption, worth up to 600 euros (Â£524).
They can also claim expenses for items such as hotels, meals and phone calls - a matter that led to a dispute with insurers during the week.
Which? said the airline sector was out of step with other service industries.
In a letter to BA chief executive Alex Cruz, Alex Neill from Which? wrote: "You have 'profusely' apologised to passengers, many of whom have had their holidays ruined, and have 'committed' to following the rules on compensation.
"This does not go far enough and is simply not good enough. You have failed your customers once and are in grave danger of doing it again.
"By simplifying the compensation process, you have an opportunity to minimise the additional stress and inconvenience you cause your customers and ensure they are not pushed into the arms of claims management companies, who will take a large part of the money they are owed.
"British Airways can, and should, seek to automatically issue statutory compensation to all affected passengers.
"This would reduce the burden on passengers and mean they get back what they are legally entitled to quicker.
"It would also allow you to focus on dealing with the individual additional expenses incurred by affected passengers on a case-by-case basis.
"Disruptions like last weekend only highlight that it is time for all airlines to introduce measures so that, where possible, passengers are compensated automatically for delays and cancellations."
A BA spokeswoman said: "We sincerely apologise for the difficulties and frustration customers faced during the huge disruption across the bank holiday weekend.
"We will fully honour our EU compensation obligations and have set up a link on the home page of our website to enable customers to submit their claims as quickly and conveniently as possible.
"We have no desire to be obstructive in any way and have put additional resources into our call centres to process claims as speedily as possible."
According to one comparison website, an estimated 70% of all airline customers who have a right to a payout do not claim.
Virgin became the first rail company to automatically compensate some train passengers if they are delayed.
Travellers using its services on the West Coast main line - and who book their tickets via the company app or website - receive automatic repayments.
Communications regulator Ofcom is also investigating the use of automatic compensation when phone or internet services fail.
At present, customers tend to go through one of two ombudsman services.
British Airways should automatically compensate thousands of customers who were caught up in disruption last weekend, says consumer group Which?