In act 4, why is Shylock considered to be a stony adversary?

Before the proceedings begin, the Duke of Venice tells Antonio that he feels sorry for Antonio, who is facing a "stony adversary" in Shylock, the unscrupulous Jewish moneylender. The Duke of Venice refers to Shylock as a "stony adversary" because he is aware that Shylock is ruthless and determined to collect his bond by any means necessary. The Duke's words ring true as he fails to persuade Shylock into exercising mercy by saving Antonio's life. After the Duke attempts to appeal to Shylock's humanity by encouraging him to have mercy on Antonio, Shylock responds by telling the Duke that he has every intention of collecting his bond and, if he cannot receive his bond, Venice's charter and freedom will be endangered.
Even Bassanio attempts to intervene and fails to persuade Shylock to spare Antonio's life. Shylock also refuses to accept double the amount of his bond when Bassanio offers him six thousand ducats, which further emphasizes Shylock's resolute nature. Portia's elegant mercy speech is also not enough to convince Shylock of sparing Antonio's life. Overall, the Duke of Venice was accurate when he referred to Shylock as a "stony adversary," and Portia is forced to cleverly interpret the laws in order to save Antonio's life.

The none too flattering description comes from the Duke of Venice ahead of Antonio's trial. The Duke is giving Antonio a warning about what kind of opponent Shylock is. When he calls him a "stony adversary," what he means is that Shylock is a hard-hearted man, someone who will prove to be a formidable opponent in court. For good measure, the Duke heaps some more choice insults on Antonio's nemesis, referring to him as "an inhuman wretch / Uncapable of pity, void and empty From any dram of mercy."
The Duke's words appear to be borne out by Shylock's subsequent behavior. For when Shylock enters the court building, he pointedly refuses to listen to the Duke's pleas for mercy on Antonio's behalf. Shylock makes it abundantly clear that he's going to enforce his bargain to the absolute letter:

I have possessed your grace of what I purpose, And by our holy Sabbath have I sworn To have the due and forfeit of my bond. (act 4, scene 1).

A "stony adversary" indeed.