Mr Mursi also promised to take steps to develop Egypt's struggling economy and to conduct foreign affairs with "dignity".
Handling relations with the Scaf is likely to be a key test for Mr Mursi as he begins his term of office.
The Scaf had previously said it would hand over power to Mr Mursi by the end of the month.
However, Scaf member Major-General Mohamed al-Assar told Egyptian media earlier this week that the head of Scaf, Field-Marshal Hussein Tantawi, would remain as defence minister under Mr Mursi.
Also on Friday, Mr Mursi performed prayers at Cairo's al-Azhar mosque, one of the most prominent seats of learning in Sunni Islam.
He has sought to allay fears among some secular and Coptic Christian Egyptians that he will use his presidency to impose Islamic law.
Mr Mursi's campaign has said he plans to appoint a woman and a Coptic Christian as his vice-presidents.