Rooney, 31, apologised after being pictured looking the worse for wear as he mingled with wedding guests at the England team hotel in the early hours during the most recent international break.
Southgate dropped him for a 0-0 World Cup qualifying draw in Slovenia, but recalled him for England's 3-0 win over Scotland, before injury forced the Manchester United forward to withdraw from the squad.
"Wayne Rooney is the England captain," Southgate told reporters at Wembley.
"I said that from the beginning of the interim period. What's also clear is that I've only selected him to start in two of the four matches we've had.
"It is not a case that Wayne expects to play every game. It's important we develop more leaders in that group.
"When I played for England at Euro 96, there were leaders throughout the team. Wayne has played an important part up to this point, but we also need to develop others."
Southgate, 46, was appointed England manager on a four-year contract on Wednesday, having previously been in charge for four games on an interim basis.
Reports suggested Southgate's contract contained a break clause after the 2018 World Cup, but he said that was not the case.
The former Middlesbrough manager was promoted from his role as England Under-21s coach after Sam Allardyce departed over a newspaper's report on his conduct.
- 'Strong and proven leader' -
Southgate's first game as full-time manager will be a friendly against Germany in Dortmund on March 22, followed by a World Cup qualifier against Lithuania at Wembley four days later.
England, humiliated by Iceland in the last 16 at Euro 2016, top their World Cup qualifying group by two points.
Several other England players were reported to have gone out on the same evening as Rooney's late night at the team hotel, prompting a review of player downtime by the Football Association.
But Southgate said he did not think he would need to remind England's players about their off-pitch responsibilities.
"I don't think laying down the law is necessary," he said.
"I like to treat players as adults. There needs to be trust between the coach and the players.
"There also needs to be clear guidelines. The players also need to take leadership on that. If we're going to have a top team, than I want them involved in that."
It is now 20 years since England last reached a major semi-final, but FA chief executive Martin Glenn predicted England will do "very well" at the next World Cup in Russia.
"He's a thinker, he's a doer," Glenn said of Southgate. "(Look at) the way he's developed and nurtured nine of the current squad, who have come directly through the under-21s.
"He's very articulate. Gareth has been a captain at every club he's played at so that says something. He's a strong and proven leader."