Mr. Trump has defended his conversation with Mr. Zelensky as “totally appropriate,” and said there had been “no quid pro quo” linking American aid to a Ukrainian investigation into Mr. Biden.

Mr. Zelensky has not announced any new investigations into Mr. Biden or his son. Speaking to reporters on Wednesday, he said Ukraine had a new prosecutor general, “a highly professional man” who would investigate “any case he considers and deems appropriate.”

“We have an independent country,” he said. “I can’t push anyone.”

Earlier on Wednesday, with the impeachment inquiry now rocketing to the center of attention in the United States, Mr. Zelensky did not make any references to Mr. Trump or even the American military aid for Ukraine in its war with Russian-backed separatists — a central element in the inquiry.

Instead, in his address to the annual General Assembly session at the United Nations, Mr. Zelensky spoke at length about the toll of wars around the world, and especially in Ukraine.

“Nobody will feel safe while Russia is waging war against Ukraine in the center of Europe,” he said. “The thought that this has nothing to do with you or will never touch your interests will be fatal.”

Mr. Zelensky also insisted that every nation had a stake in conflicts that seemed distant.

“We cannot think globally while turning a blind eye to small things, or as some may believe, to trifles,” he continued. “That is how the foundation of two world wars was laid down, and as a result millions of human lives have paid the price for negligence, silence, inaction or an unwillingness to relinquish our own ambitions.”

Mr. Zelensky, 41, vaulted into Ukraine’s highest office this year with an unconventional campaign and an even more unconventional background: He starred in a popular TV comedy about a schoolteacher who is unexpectedly elected president after a rant about corruption is posted online.