Actually papal consent was no longer required. On July 16, 1338, six of the seven electors met at Rhense and declared that election by a majority of electors was enough to make someone King of the Romans and future emperor, and popes had no right to interfere. Emperor Louis IV decreed on August 6, 1338, that the person elected immediately had full imperial rights and powers, and that everyone was obligated to ignore any papal efforts to undermine and oppose the Emperor-elect.
No emperor since Charles V in 1530 had been crowned by the pope. Instead they took the title of Emperor elect upon being crowned in Germany.
If Gustavus Adolphus wanted to make himself Emperor of the Romans he would have had to get a majority of the seven electors to declare Emperor Ferdinand II deposed and himself elected. The electors in 1630-1632 included two, the Margrave of Brandenburg and the Duke of Saxe-Wittenburg, who were Protestants and likely - but not certain - to vote for Gustavus Adolphus as emperor.
Two other electorates, the Kingdom of Bohemia and the Palatinate of the Rhine, had recently had Protestant rulers. Bohemian dissidents had elected Frederick V, Count Palatine of the Rhine, king in 1619. The rightful king, Emperor Ferdinand II, and the Catholic League led by Duke Maximilian of Bavaria, had invaded Bohemia and driven Frederick out in 1620. They then conquered the Palatinate in 1621-1622. In 1623 Emperor Ferdinand granted the electorship and some of its lands to Duke Maximilian of Bavaria, leader of the Catholic League. Emperor Ferdinand made his son, King Ferdinand of Hungary, king of Bohemia in 1627. The younger Ferdinand was elected King of the Romans in 1636 and became emperor in 1637.
So in 1630-1632 two of the electors would have been likely to vote for Gustavus Adolphus, two were among his biggest enemies, and the other three were the Catholic Archbishops of Mainz, Trier, and Cologne, naturally unlikely to vote for the Protestant champion.
If Gustavus Adolphus had been able to occupy Bohemia and Bavaria, he might have been able to restore Frederick to the Bohemian throne and electorship and force Maximilian to return his electorship to Frederick. Gustavus could have occupied one or more of the three electorates of Mainz, Trier, and Cologne and forced their electors to agree to vote for him in return for the restoration of their realms.
So Gustavus Adolphus could have assembled enough votes to get elected emperor if he had won more battles and occupied more territory. And he could have had a group of Protestant leaders proclaim him emperor without a meeting of the seven electors if he had won enough battles and done enough to protect and spread Protestantism.
But the more forcefully and unconstitutionally Gustavus Adolphus acted the less he would seem like a defender of German freedom against the "tyrannical" Ferdinand II and the more he would seem like a more tyrannical and dangerous alternative. So his political and military and propaganda options to become emperor were uncertain.
Of course Gustavus Adolphus could have proclaimed himself himself Great King of Sweden or High King of Sweden or King of Kings of Sweden and it might have been accepted by his allies, though not by his enemies. But if he proclaimed himself Emperor of the Swedish Empire even his allies might have laughed at the idea that there could be another empire beside the Roman Empire.