Salvo, a local businessman backed by the club's foundation and major shareholder, pledged "democratisation, union and excitement" after winning more than 90 percent of the votes at a meeting late on Tuesday.
"If there is no union we will not be able to refinance our debts, we will not finish building the new stadium and we won't be able to bring players from the academy into the first team," Salvo said.
Valencia also confirmed that new coach Miroslav Djukic had agreed a two-year contract as he takes over from the departing Ernesto Valverde.
Djukic, a former Serbia coach who spent six seasons with Valencia as a player, will oversee the club's Europa League, La Liga and King's Cup campaigns next season.
Valverde, brought in to replace Mauricio Pellegrino in December on a six-month contract, said institutional instability had convinced him it was best to leave.
Valencia lost 4-3 at Sevilla on the final day of the La Liga season on Saturday and missed out on a Champions League place.