The European Union Aviation Safety Agency on Tuesday said it "is taking every step necessary to ensure the safety of passengers." The decision not only applies to airlines within the European Union but to operators outside of the region flying to or from the region, the regulator said.

Aviation regulators from China to Britain have grounded the aircraft, joining a growing list of countries and airlines suspending the plane and banning it from their airspace after the second deadly crash of the popular aircraft in less than five months.

Europe's aviation regulator on Tuesday said it is suspending operations of Boeing 737 Max 8 jets following the deadly crash of one of the planes in Ethiopia over the weekend — leaving the U.S. increasingly alone in standing by the American-made aircraft.

Boeing 737 Max-8 aircraft are parked up at a gate in the terminal of Manchester Airport on March 12, 2019 in Manchester, England.

The Federal Aviation Administration said Monday that it did not see a reason to ground the planes and Boeing said it wasn't planning to issue new guidance to pilots "based on the information currently available."

"We understand that regulatory agencies and customers have made decisions that they believe are most appropriate for their home markets," Boeing said in a statement. "We'll continue to engage with all of them to ensure they have all the information they need to have the confidence they need safely continue to operate their fleets or return them to service."

Boeing shares were down more than 6 percent in afternoon trading Tuesday.

The crash of Ethiopian Airlines Flight 302 shortly after takeoff from Addis Ababa on Sunday killed all 157 people on board. Investigators have recovered the two black boxes from the crash site, which should provide information about what brought down the Nairobi, Kenya-bound aircraft. The crash happened less than five months after a Lion Air Boeing 737 Max 8 went down in Indonesia in October, killing all 157 people on board.

American Airlines and Southwest Airlines flight attendants and ground crews urged the companies to take their Boeing 737 Max airplanes out of service.

"Our Flight Attendants are very concerned with the recent Ethiopian Airlines flight 302 crash, which has raised safety concerns with the 737 MAX 8," said Lori Bassani, president of the Association of Professional Flight Attendants, which represents American's some 27,000 flight attendants. "Many respected global carriers are grounding the planes. We are calling on our CEO Doug Parker to strongly consider grounding these planes until a thorough investigation can be performed."

American and Southwest on Tuesday told CNBC that they still have confidence in the aircraft and their crews.

Southwest has 34 Boeing 737 Max 8 planes in its fleet and said it would "waive fare differences that might normally apply" for travelers who wish to "rebook their flight to another aircraft type," spokesman Dan Landson told CNBC. The planes represent a small number of Southwest's fleet of more than 750 Boeing 737s, which is mostly models older than the Max.

American said it has not lifted its ticket change fees, which can cost $200 or more, for travelers who want to avoid the Boeing 737 Max.

"Our flight, flight service, tech ops and our safety teams, along with the heads of our pilot and flight attendant unions, will continue to closely monitor the accident investigation from this weekend," American said in a statement. "We are keen to learn any findings from the cockpit voice recorder and flight data recorder, which will provide a better understanding of the cause of this recent accident."

Flydubai said Tuesday that it's taking its 11 Boeing 737 Max 8 and two Boeing 737 Max 9 planes out of service after the United Arab Emirates aviation authority grounded the jets.

— CNBC's Jessica Bursztynsky and Phil LeBeau contributed to this article.