Even though you don't think you're moving your eyes when looking at clouds and tree bark, you more then likely are, just a small amount, and at hard to notice speeds. You're so used to this process that its not something you can be sure that you would notice, since you normally don't. You're brain combines all these part images together to give you what you perceive as "the world." It is also very possible that you aren't truly seeing the bark. You've seen it many times before, even if you don't realize it. You're brain will often fill in information that it isn't actually "seeing" at that very moment because it either knows that it is there, or that it believes that it is there. This is one of the reasons why eye witness accounts of sudden and traumatic crimes can be notoriously inaccurate. As an example, a person may honestly believe that a mugger has a gun in his one hand that is down at his side near his (the robbers) pocket, but in reality what is there is a dark pattern on his jacket pocket, either from the jacket's colour/style, or from a shadow. The witness isn't lying, he was just so afraid for his/her life that he/she imagined that there was a gun there.
Probably the only real, reliable way to conduct an experiment like looking at a very dark and a very bright thing at the same time and knowing that you didn't look at each separately would be to have a special camera/s closely monitoring your head, eyeballs, and pupils for any movement. It would also have to be an artificial or set up scene so that there was some symbol or something in the dark area that you would have to be able to identify without any movement. I honestly don't think "not moving" at all is possible without medical intervention such as somehow disabling a persons ability to move at all; body, neck, head, even eyeballs.
Not exactly a fun experiment.