Debris fell on the field and main grandstand, narrowly missing France coach Didier Deschamps, when the parachutist struck wires for an overhead camera attached to the stadium roof.
UEFA called it a "reckless and dangerous'' act and said "law authorities will take the necessary action.''
"This inconsiderate act ... caused injuries to several people attending the game who are now in hospital,'' UEFA said.
The incident happened just before the start of the Euro 2020 match between the last two World Cup champions. Deschamps was shown ducking into the team dugout to avoid falling debris.
"We as the German football federation condemn it of course, because it wasn't just him, but others that he endangered and injured. It's unacceptable from our point of view," German team spokesman Jens Grittner said. "And the incident is being checked by the police, the authorities here in Munich and at UEFA. But of course we also condemn what happened there. It could probably have turned out much worse.''
The protestor's parachute had the slogan "KICK OUT OIL!" and "Greenpeace'' written on it. He glided into the stadium and seemed to lose control after connecting with the wires. He veered away from the playing area toward the main grandstand and barely cleared the heads of spectators.
The parachutist managed to land on the field, and Germany players Antonio Rudiger and Robin Gosens were the first to approach him. He was led away by security stewards and given medical attention on the side of the field.
UEFA and one of its top-tier tournament sponsors, Russian state energy firm Gazprom, have previously been targeted by Greenpeace protests. In 2013, a Champions League game at FC Basel was disrupted when Greenpeace activists abseiled from the roof of the stadium to unfurl a banner protesting Russian oil and Gazprom, which sponsored the visiting team, German club Schalke 04.
Greenpeace later donated money to a charity supported by Basel, which was fined by UEFA for the security lapse.
UEFA defended its environmental credentials in Tuesday's statement.
"UEFA and its partners are fully committed to a sustainable Euro 2020 tournament," UEFA said, "and many initiatives have been implemented to offset carbon emissions.''