BEIJING—The leaders of China and Taiwan sparred over the prospects of unification, showing the decades-old split remains potentially volatile under intensifying pressure from Chinese President Xi Jinping to change the status quo.

In a tightly calibrated 33-minute speech meant as an address to Taiwan on Wednesday, Mr. Xi said differences in political systems can’t be used as an excuse to resist unification. He promised Taiwanese people a peaceful and prosperous future with the mainland while suggesting that Beijing’s patience...