Wenger, 67, announced he will step down from his role at the end of the season on Friday after 22 years in charge.
Despite winning three Premier League titles and a record seven FA Cups, mounting pressure was put on Wenger to leave by the club's disgruntled fans as he failed to match the standards he set early in his reign over the past decade.
The Gunners haven't won the Premier League for 14 years and will miss out on Champions League qualification for the second successive season unless they win the Europa League, where they face Spanish giants Atletico Madrid in the semi-finals.
"We are not going to find a replacement for Arsene Wenger for a variety of reasons," said Gazidis as he addressed a press conference at the club's Emirates stadium on Friday.
"The football club is in a very different place than it was 22 years ago and it's unthinkable to me we will have another manager in the Premier League, let alone Arsenal, who will be 22 years in tenure and have the kind of run of consistent success Arsene had over those years."
In spite of Arsenal's struggles on the field in recent seasons, they remain the sixth richest club in the world, according to financial consultants Deloitte's annual Football Money League.
And Gazidis claimed Wenger leaves the club in far better shape than when he arrived as a shock appointment from Japanese side Grampus Eight in 1996.
"Arsene often said he wanted to leave the club in a better position than when he found it," added Gazidis.
"We are in a better place today than we could ever have imagined 22 years ago. The foundations of this club have never been stronger and this gives me great confidence as we begin to chart the path ahead."
However, Gazidis repeatedly avoided questioning on whether he wanted Wenger to see out the final year of a two-year contract extension he signed last summer.
The Wall Street Journal reported that Wenger had chosen to leave on his own terms rather than face being pushed out at the end of the season.
Gazidis, though, claimed the wheels hadn't been put in motion to appoint a successor until Wenger made his decision public.
Former Arsenal captain Patrick Vieira, Celtic manager Brendan Rodgers and Germany coach Joachim Loew are among the front runners with the bookies in a wide-open field of contenders.
"We haven't had any discussions to date regarding that process," added Gazidis.
"The most important thing is to make the right appointment not a quick appointment."