The Federal Aviation Administration said Monday it doesn’t plan to ground Boeing Co.’s 737 MAX planes, despite concerns by other countries, passengers and airline employees after the model’s second crash in less than five months.

“This investigation has just begun and to date we have not been provided data to draw any conclusions or take any actions,” the FAA said in an update to airlines, referring to Sunday’s deadly Ethiopian Airlines crash. The agency said it expected by the end of April to mandate a software fix for an...