Myanmar’s military, fresh off a bloody campaign that drove most of the country’s Muslim Rohingya minority into neighboring Bangladesh, has launched a new offensive against the armed forces of an ethnic group along the northern border with China using helicopter gunships, jet fighters and heavy artillery.

The campaign against the Kachin, a mostly Christian minority in Myanmar’s remote northern mountains, reflects an accelerated effort to pummel recalcitrant militias that refuse to sign on to a national cease-fire agreement...