SEOUL, South Korea — A day after exchanging artillery fire across their disputed sea border, North and South Korea hurled insults at each other on Tuesday, with the North rejecting an ambitious overture from the South’s president, Park Geun-hye.

In a speech in Dresden, Germany, that was broadcast live in South Korea on Friday, Ms. Park promised a huge investment in North Korea’s decrepit industries, as well as humanitarian aid for babies and nursing mothers, if the North gave up its nuclear weapons program.

Although Ms. Park’s proposal largely reiterated what her three predecessors had proposed, it was her most detailed overture toward the North since she came to office shortly after the North conducted a nuclear test in February 2013.

On Tuesday, the state-controlled North Korean newspaper Rodong Sinmun scoffed at Ms. Park’s speech, calling the unmarried South Korean leader an “eccentric old spinster” and “a frog in a well.” It said her overture was “full of deception” and “filth” and was aimed at destroying the North Korean government.