The 46-year-old was promoted from his role with the England Under-21s last week after Sam Allardyce left his post following a newspaper sting.
Southgate ruled himself out of the running in the summer following Roy Hodgson's resignation but refused to comment on if he has changed his mind now he will manager England's next set of World Cup qualifiers.
"The last week has been very difficult," he said. "We have got an opportunity to work with a group of players that are young and need guidance and stability.
"I have not had any chance to think about anything other than this week and getting the best results we want. Anything beyond that can wait.
"The squad in Slovakia did very well, why veer too much from that? We lost a few players who need replacements but I felt stability was important."
But the former Middlesbrough manager couldn't hide his pride and is looking forward to the challenge.
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"The role is an honour. It was a difficult position for the organisation and it was important to show leadership and get things in place to give the team the chance of winning," Southgate added.
"As the week goes on I have thought about the honour of what it is to lead the team. I am looking forward to the challenge, in my view it is a wonderful job to have for whatever period of time.
"It is huge for my family and huge for everyone who has helped me in my career, everyone who has touched my career and helped me as a person will take enormous pride in seeing me take charge of the team."