The disease associated with low cortisol is known as adrenal insufficiency (so-called because the adrenal glands are responsible for cortisol production). Morning cortisol is not a definitive test for the diagnosis of adrenal insufficiency.1 That is because cortisol varies according to a circadian rhythm (i.e. with your body's sleep-wake cycle).
The x axis represents time across a 24h day.2 The transition from dark yellow to light yellow indicates the habitual wake time. The y axis shows plasma cortisol level.
The OP mentioned in comments that this value was obtained four hours prior to his habitual wake time. As shown on the graph above, this is predictably a trough (i.e. low point) in serum cortisol. On the other hand, the "normal" range is likely calibrated for the peak level. (This test is supposed to be done shortly after waking, approximating the peak.) The fact that a value below the normal range was obtained is predictable.
For this reason, a confirmatory test is required to make the diagnosis of adrenal insufficiency. This generally involves administration of ACTH (adrenocorticotropic hormone) – the physiologic stimulator of cortisol production – and assaying cortisol levels at fixed intervals afterward to evaluate for an adequate rise. Only after inadequate adrenal response is demonstrated can the diagnosis be made.3
In direct response to the OP's questions, then:
It is normal to treat low cortisol levels....?
No, except in exceptional circumstances, a confirmatory test is required before treatment.
It is normal to treat...without symptoms?
The symptoms of adrenal insufficiency are notoriously vague, and the OP indicates that there actually was some complaint prompting the testing. Inadequate treatment of adrenal insufficiency can go unnoticed until a stressor triggers a life-threatening condition known as adrenal crisis. If a true inadequate response to ACTH is demonstrated, this should be treated.
Is it normal to not do it any further screening of possible causes (my doctor didn't)?
It is standard to follow a low cortisol level with a ACTH stimulation test. On the other hand, if this is only mildly below the reference range of your lab, given the likelihood that what has actually been measured is closer to a trough than a peak cortisol level, one could imagine a scenario where it might be reasonable to drop it. In addition to the reference range, the details of the presenting complaint and any physical exam or laboratory abnormalities would be necessary to make a decision in that regard. This part is beyond the scope of Health.SE.
1. A normal value, on the other hand, can rule out the condition.
2. Note that the numbers are arbitrary; circadian time is not equal to clock time even if they do correspond in an individual accustomed to waking at 0800.
3. As with most things in life, it's actually more complicated. It's possible to have secondary adrenal insufficiency (where the problem involves the brain's production of ACTH) and still "pass" an ACTH stim. test if the condition is of recent onset and the adrenals have not yet atrophied.
Nicola Neary and Lynnette Nieman. Adrenal Insufficiency- etiology, diagnosis and treatment Curr Opin Endocrinol Diabetes Obes. 2010 Jun; 17(3): 217–223.
Mark S. Rea et al. Relationship of Morning Cortisol to Circadian Phase and Rising Time in Young Adults with Delayed Sleep Times International Journal of Endocrinology, 2012.