BUG: Fix read_json orient='table' without index (#25170)

pandas/io/json/table_schema.py

@@ -314,12 +314,13 @@ def parse_table_schema(json, precise_float):
 
     df = df.astype(dtypes)
 
-    df = df.set_index(table['schema']['primaryKey'])
-    if len(df.index.names) == 1:
-        if df.index.name == 'index':
-            df.index.name = None
-    else:
-        df.index.names = [None if x.startswith('level_') else x for x in
-                          df.index.names]
+    if 'primaryKey' in table['schema']:
+        df = df.set_index(table['schema']['primaryKey'])
+        if len(df.index.names) == 1:
+            if df.index.name == 'index':
+                df.index.name = None
+        else:
+            df.index.names = [None if x.startswith('level_') else x for x in
+                              df.index.names]
 
     return df

pandas/tests/io/json/test_pandas.py

@@ -1262,3 +1262,12 @@ def test_index_false_error_to_json(self, orient):
                "'orient' is 'split' or 'table'")
         with pytest.raises(ValueError, match=msg):
             df.to_json(orient=orient, index=False)
+
+    @pytest.mark.parametrize('orient', ['split', 'table'])
+    def test_index_false_from_json_to_json(self, orient):
+        # GH25170
+        # Test index=False in from_json to_json
+        df = DataFrame({'a': [1, 2], 'b': [3, 4]})
+        dfjson = df.to_json(orient=orient, index=False)
+        result = read_json(dfjson, orient=orient)
+        assert_frame_equal(result, df)