"Our project is to form a stable progressive government led by the @PSOE to make policies for the benefit of the majority," acting Prime Minister and party leader Pedro Sanchez said in a tweet. "All Parties must act with generosity and responsibility starting tomorrow to unlock the country."
PSOE won the most seats in elections in April but, after months of negotiations, failed to form a government.
The conservative Popular Party (PP) won the second-most seats on Sunday with 87. Vox made major gains, seeing its seat count rise from 24 to 52. A party needs at least 176 seats in the Congress in order to gain control of the government.
Spain has been in political stalemate since former Prime Minister Mariano Rajoy was ousted by a no-confidence vote over a corruption scandal in his party in June 2018.
The country's two-party system has also fragmented in recent years, with both the PSOE and the PP coming under pressure following the 2008 financial crisis and a series of corruption scandals.
The emergence of the left-wing Unidos Podemos (UP) and the center-right Ciudadanos (Citizens) party, and the growth of Vox, also shook up the established political order. A number of groups representing Spain's autonomous regions, such as Catalonia and the Basque Country, have also risen in prominence.
The election was called after Sanchez failed to secure enough support in the 350-seat Parliament to form a government, despite months of negotiations. He announced in September that Spain would go to the polls November 10, less than seven months after the April general election that proved inconclusive.
"There is no majority in congress that guarantees the formation of a government," said Sanchez.
Blaming opposition parties for the political deadlock, Sanchez added: "I've done all I can, but they've made it impossible."
Spain is the only western European country never to have been governed by a coalition.
In recent years, several minority governments have been shored up by parliamentary alliances, but with a number of polarizing issues in play this time around, forming a government has been difficult.
The current political crisis, which is the worst the country has faced since the restoration of democracy in the late 1970s, is tied to the issue of Catalan independence.
While the Socialist party has won votes in urban areas thanks to progressive reforms on women's and LGBT rights, the pace of change has also led to a backlash in other places.
CNN's Jack Guy and Aimee Lewis contributed to this report.