A fourth-place finish in 2018-19 was not enough to keep Spalletti in the job as Inter turned to former Juventus boss Conte on a three-year contract.
Conte has been out of management since leaving Stamford Bridge in 2018. His first season in London saw the Blues claim the 2016-2017 Premier League title, but uneven performances the next year saw the club miss out on qualification for the Champions League.
"Spalletti is a great person and a great coach, but the club made its decision," he told Sport Mediaset. "Now there is Conte, there are new expectations and we want to challenge for the Scudetto.
"Last season was satisfactory because we got into the Champions League, but we can always do better.
"The results in the final weeks of the season weren't what we wanted, but our objective was to finish in the top four and we achieved that."
Inter have been tipped to bring in a number of new players ahead of Conte's first season at San Siro, with experienced Uruguay international centre-back Diego Godin widely expected to be on his way from Atletico Madrid on a free transfer, but prolific striker Mauro Icardi's future remains unclear.
"Icardi is a great player and a friend, so we'll see what happens. If he remains here, then we'll be happy about that," Skriniar said.
"Godin will make the squad even more competitive and that is the most important thing."
Manchester United striker Romelu Lukaku has been strongly linked with a move to Inter, but Icardi was not featured in the club’s recent kit launch, with some taking that as a sign he’ll leave San Siro.