MOSCOW—The Kremlin stated it experienced no involvement in the unexpected health issues of Russian dissident Alexei Navalny and questioned the assertion by German health professionals that he was poisoned, pushing back against needs from Western leaders for an inquiry into the most current suspected attack against a critic of President Vladimir Putin.

The exact situation encompassing the health issues of Mr. Navalny, who continues to be in a coma in a Berlin clinic, are a mystery. But Kremlin spokesman Dmitry Peskov explained to journalists Tuesday that ideas that Mr….