WASHINGTON—President Donald Trump, surrounded by addicts and their families, declared opioid addiction a “public health emergency” Thursday as he sought to accelerate a federal government response to the crisis.

“We can be the generation that ends the opioid epidemic,” Mr. Trump said in a 25-minute speech from the White House in which he decried the tragedy of “opioid orphans” and described opioid abuse as a “plague” that Americans must defeat.

Mr....