WASHINGTON—President Barack Obama's package of gun-control measures immediately reignited longstanding divisions between liberal members of his own party eager to identify with a clampdown on guns and more-centrist Democrats who pride themselves on defending gun rights, underscoring the fraught path ahead.

Girding for resistance, the president in his Wednesday address called for people across the country to lobby their lawmakers to overcome certain opposition from the National Rifle Association, the nation's largest gun-rights...