1.) Right click on the word "healed" in Isaiah 53 and select the Hebrew lemma (this will find all of the grammatical forms of that word, no matter how they are translated in English) on the right click menu.
2.) Left click on "Bible Word Study" in the menu that popped up. 
2.) Left click on "Bible Word Study" in the menu that popped up. 
3.) Scroll down to "Clause Participants" on the Bible Word Study window (although you may find the pie chart of the ways this verse is translated helpful). Make sure "Grammatical Roles" is selected under clause participants (you can also use semantic roles and use "agent" instead of subject, but I am using grammatical for this example).
4.) Click on the down arrow to see all of the places where God is the subject of healed.
5.) You will notice that God is the subject of this lemma for healed 25 times (Isaiah 53 is not on this list, because the Righteous Servant is the subject, not God). Almost all of these refer to God physically healing someone (or the land, or a spring). But the exceptions are notable: Isaiah 57:18-19, Jeremiah 3:22, Jeremiah 17:14, Psalm 41:5 and Psalm 147. Hosea 11:3 may be debatable, but I see it as spiritual. So, strictly on proportions, we would see the healing as physical. But the places where it is clearly spiritual refer to the restoration of peace and the healing of the brokenhearted. That inclines me to think it may be spiritual here.
6.) If we wanted to do more, I would right click on "healing" again, select the English word this time and do an English word study to see if there is a Hebrew word which usually means "heal" metaphorically, or if there is another word which always means "heal" literally. I haven't done this study, but honestly do not expect it to be fruitful. But if you are serious about the topic, it is necessary to check.
7.) Then I would check cross references. Again, Logos makes this very easy with the Treasury of Scripture Knowledge. Many of the cross references just go to further reinforce the idea that healing can be spiritual or physical. If we were left here, we might have to just try and reason it out from the context. Is Isaiah adding something here about physical healing, or is he repeating himself about the idea of spiritual healing? (On that count, the general rule of hermeneutics is that, when there are multiple possibilities, the best interpretation of a word is the one which adds the least to the passage, which would mean spiritual healing would fit here the best. It may seem like a strange rule, but it is basically just a clear test for what fits the context.).
8.) One cross reference in particular is important here, though. Matthew 8:17 quotes (sort of, the way NT authors paraphrase the OT is a study for another day) this passage. So what is the inspired interpretation? From the ESV: "Matthew 8:14–17 (ESV)
Jesus Heals Many
14 And when Jesus entered Peter’s house, he saw his mother-in-law lying sick with a fever. 15 He touched her hand, and the fever left her, and she rose and began to serve him. 16 That evening they brought to him many who were oppressed by demons, and he cast out the spirits with a word and healed all who were sick. 17 This was to fulfill what was spoken by the prophet Isaiah: “He took our illnesses and bore our diseases.”"
Since the actual quoted verse says "He has borne our griefs and carried our sorrows," Matthew's paraphrase here is a way of using the beginning of a passage to summarize its contents. Matthew clearly sees physical healing present in Isaiah 53, and verse 5 is the only place it could conceivably fit in. While he also sees spiritual healing in the casting out of demons, this is secondary in the context.
So physical or spiritual healing? The answer must be, from the immediate context, the regular meaning of the word and the New Testament: both!
