Did the Holy Prophet Muhammad (saw) also greet Jews with the greeting of peace?

Yes. He greeted people of every religion with the greeting of peace. Only when someone twisted the greeting to wish him harm did he reply briefly with "and upon you".

In Medina, Muslims, Jews and followers of other beliefs lived side by side. The greeting of peace, "as-salamu alaikum", peace be upon you, was part of daily life there. The Holy Prophet Muhammad (saw) used it with all people, not only with Muslims.

The first hadith above reports that he passed by a group made up of Muslims, idol worshippers and Jews, and greeted them all with the greeting of peace. People of other religions who came to him also addressed him with good wishes.

Some, however, misused the greeting. They spoke so indistinctly that it seemed to sound like "as-salamu alaikum", but in truth they said "as-samu alaikum", death be upon you. In this way the wish for peace secretly became a curse. The second hadith shows how the Holy Prophet Muhammad (saw) responded. He said only "and upon you". When someone near him answered back with harsh words, he urged gentleness, because Allah loves gentleness in every matter.

So the short reply was not a different greeting for Jews. It was a measured response to a twisted greeting. The principle remains what the verse above says. A greeting is returned with a better one, or at least an equal one.

The lesson for today follows from this. Muslims greet people of other religions with the greeting of peace and return their sincere greetings warmly.