The regulator said the airlines faced unlimited fines if they failed to agree to pay up and a court order was taken out against them.
However, Etihad said the CAA’s announcement was “unprofessional and unacceptable” because it said discussions were continuing.
Emirates described it as "misleading" and "absurd".
The carriers carry tens of thousands of Scottish passengers a year to long-haul destinations via their hub airports.
Three operate from Edinburgh, with American flying to New York, Etihad to Abu Dhabi and Turkish to Istanbul.
Emirates flies twice a day between Glasgow and Dubai, while other Scottish travellers connect with Singapore flights in London and Manchester.
The CAA said the airlines had breached consumer law by not compensating passengers who missed a connecting flight because the first leg of their flight was delayed and arrived more than three hours late.
The regulator said more than 200,000 passengers a year could be affected.
It said Emirates was the complained-about airline for not paying compensation for connecting flights.
The CAA said if the airlines did not sign a legal undertaking, they could be found in contempt of court.
Director of consumers and markets Richard Moriarty said: “Delays that cause people to miss connecting flights have a particularly damaging effect on people’s travel plans.
“Airlines’ first responsibility should be looking after their passengers, not finding ways in which they can prevent passengers upholding their rights.
“So it’s disappointing to see a small number of airlines continuing to let a number of their passengers down by refusing to pay them the compensation they are entitled to.”
Alex Neill, home and legal services managing director for consumer group Which?, said: “It is appalling that some airlines believe they are above the law and do not have to pay passengers the compensation they are legally entitled to.
“It should be much simpler for passengers to claim money for delays and the [UK] Government must legislate for a new ombudsman that all airlines are required to join.”
However, three of the airlines disagreed with the CAA’s decision.
An Etihad spokesman said: “We have been engaged in constructive dialogue with the CAA on the issue of passenger compensation over several months.
“We take such matters very seriously and in no way look to breach the law.
“Therefore, before even completing the dialogue, we find the CAA’s approach wholly ‘unprofessional and unacceptable’ to publicly blame Etihad Airways for infringements to passengers’ rights, which we unreservedly deny.
“Further to our own review, we categorically state we have not contravened the law, yet pledge to maintain dialogue with the CAA and regulators to ensure we continue to have robust compensation policies in place.
“The issue of a non-EU airlines’ liability under EC261 for a delay due to a missed connecting flight outside of the EU is currently subject to a case that will be heard by the UK Court of Appeal.
“Etihad Airways will, of course, abide by any decision of the Court of Appeal.”
“As one of the world’s largest airlines, we comply with all legal requirements and regulations as set by the relevant authorities.
Emirates also attacked the CAA move.
A spokesman said: “The way in which the CAA has communicated this issue is both misleading and unprofessional.
"As the CAA is well aware, the recent EU guidelines on EC 261 are not intended to amend the law.
"The issue of EC 261’s application to our flights from the UK involving a stopover in Dubai is currently pending before the Court of Appeal.
"We will rigorously defend our position, and challenge the blanket application of EC 261 to every situation, without consideration of context or the safety of our passengers.
Emirates, like any responsible airline, puts the safety of our passengers first and to be penalised for this is absurd.
A spokeswoman for American Airlines said: “We are aware of the connecting flight portion of the Civil Aviation Authority report, which relates to a discrete legal issue involving EC261.
“We disagree with the CAA’s interpretation of this legal issue and look forward to additional conversations on the matter.”