The answer is "No". The Vatican did not build a secret property empire, they did this openly, and they did use the millions they received from Mussolini.
In this official Vatican reaction in the Telegraph.co.uk to the secret property article in the Guardian.co.uk, the Vatican spokesman Father Federico Lombardi explaines:
The money was compensation, issued under the Lateran Accords of
1929, for the properties the Vatican lost in the Papal States,
such as the Quirinal Palace in Rome, which has since become the
residence of the Italian President.
"This article reveals nothing that was not known already," he
commented. "The fact that Administration of the Patrimony of the Holy
See has a special section [for real estate] is even recorded in the
Vatican telephone exchange."
So the money was received by the Italian state, which was under control by the fascist regime of Mussolini at the time. But, there was a lot more than just money involved in this deal. This deal made the Vatican an official country again, catholicism the official Italian religion and much more. Here is a great article about "How the Lateran Treaty made the Catholic Church into a state"
Similar deals (Concordats) were made around the world between countries and the Vatican. The Italian Lateran Treaty (1929) was controversial, but even more so was the deal with Germany in 1933 with the Nazi regime known as the Reichskonkordat (1933)
Both deals pretty much gave moral support and legitimacy to the fascists party in Europe in the 1930's and all the horrors that happened the following 15 years. Though, never official of course, but pope Pius XII's (and the Vatican's) role during, and the years leading up to, WWII remains controversial.