Is there a non surgical way to fix downturned eyes? (Photo)

There is no effective long term treatment for downturned eyes other than surgery. See following link on details about almond eye surgery. 

So simply speaking, you cannot improve your downturned lids without surgery. There are some fairly simple surgical procedures under local anaesthetic that can help that don't take vey long and have minimal downtime so don't write surgery off completely. Ask a local surgeon what they think and go from there. Hope that helps. Adam Goodwin

Thank you for including a high-quality photograph.  In reality, the only way you will obtain significant change in the position of your lower eyelids is to undergo a surgical procedure.  It would be relatively straightforward and predictable, and would improve the overall aesthetics of your already lovely eyes.  You may also benefit from fat transfer or filler injection as well to help provide some support to the lower eyelid procedure.  Best wishes for a beautiful result!