BUG: Do not use string Index like Datetimelike Index

pandas/core/indexes/datetimelike.py

@@ -138,13 +138,16 @@ def equals(self, other) -> bool:
         if not isinstance(other, ABCIndexClass):
             return False
         elif not isinstance(other, type(self)):
-            try:
-                other = type(self)(other)
-            except (ValueError, TypeError, OverflowError):
-                # e.g.
-                #  ValueError -> cannot parse str entry, or OutOfBoundsDatetime
-                #  TypeError  -> trying to convert IntervalIndex to DatetimeIndex
-                #  OverflowError -> Index([very_large_timedeltas])
+            if self._is_convertible_to_index_for_join(other):
+                try:
+                    other = type(self)(other)
+                except (ValueError, TypeError, OverflowError):
+                    # e.g.
+                    #  ValueError -> cannot parse str entry, or OutOfBoundsDatetime
+                    #  TypeError  -> trying to convert IntervalIndex to DatetimeIndex
+                    #  OverflowError -> Index([very_large_timedeltas])
+                    return False
+            else:
                 return False
 
         if not is_dtype_equal(self.dtype, other.dtype):
@@ -596,6 +599,26 @@ def _convert_arr_indexer(self, keyarr):
             converted_arr = com.asarray_tuplesafe(keyarr)
         return converted_arr
 
+    @classmethod
+    def _is_convertible_to_index_for_join(cls, other: Index) -> bool:
+        """
+        return a boolean whether I can attempt conversion to a
+        DatetimeIndex/TimedeltaIndex
+        """
+        if isinstance(other, cls):
+            return False
+        elif len(other) > 0 and other.inferred_type not in (
+            "floating",
+            "mixed-integer",
+            "integer",
+            "integer-na",
+            "mixed-integer-float",
+            "mixed",
+            "string",
+        ):
+            return True
+        return False
+
 
 class DatetimeTimedeltaMixin(DatetimeIndexOpsMixin, Int64Index):
     """
@@ -903,6 +926,15 @@ def _is_convertible_to_index_for_join(cls, other: Index) -> bool:
             return True
         return False
 
+    def _wrap_joined_index(self, joined: np.ndarray, other):
+        assert other.dtype == self.dtype, (other.dtype, self.dtype)
+        name = get_op_result_name(self, other)
+
+        freq = self.freq if self._can_fast_union(other) else None
+        new_data = type(self._data)._simple_new(joined, dtype=self.dtype, freq=freq)
+
+        return type(self)._simple_new(new_data, name=name)
+
     # --------------------------------------------------------------------
     # List-Like Methods
 

pandas/tests/indexes/datetimes/test_misc.py

@@ -408,3 +408,20 @@ def test_isocalendar_returns_correct_values_close_to_new_year_with_tz():
         dtype="UInt32",
     )
     tm.assert_frame_equal(result, expected_data_frame)
+
+
+def test_datetimelike_string():
+    # Related to PR 32739
+    # Ensure we do not compare strings and datetimelike type.
+    date_string = "2020-04-13"
+    i1 = pd.Index([date_string])
+    i2 = pd.Index([pd.to_datetime(date_string)])
+
+    assert i1.equals(i2) is False
+    assert i2.equals(i1) is False
+
+    assert len(i1.intersection(i2)) == 0
+    assert len(i2.intersection(i1)) == 0
+
+    assert len(i1.union(i2)) == 2
+    assert len(i2.union(i1)) == 2