47 CFR 80.148 states: "Each compulsory vessel, while underway, must maintain a watch for radiotelephone distress calls on 156.800 MHz [some exceptions follow]..."
http://www.gpo.gov/fdsys/pkg/CFR-2011-title47-vol5/pdf/CFR-2011-title47-vol5-sec80-148.pdf
Note the qualifier "compulsory," meaning: "the vessel is required to have a radio."
According to that rule, if your vessel is not required to have a radio, the radio is not required to be turned on and tuned to channel 16.
However, 47 CFR § 80.310 contradicts, saying you MUST maintain a radio watch if you have a radio, even when your vessel is "voluntarily" equipped with a radio. In the case of vessels having limited battery power, I believe it is perfectly legitimate to say that your radio was not on to conserve its batteries for possible emergency need.
http://www.gpo.gov/fdsys/pkg/CFR-2011-title47-vol5/pdf/CFR-2011-title47-vol5-sec80-310.pdf
That rule also states a requirement to monitor 2182 kHz, even though many vessels lack the ability to receive that frequency and even the US Coast Guard has itself stopped monitoring the frequency - a case of the rules not keeping up with the times.
http://alaska.coastguard.dodlive.mi...ast-guard-ceases-monitoring-channel-2182-khz/
I have ample solar power generating ability aboard my sailboat, and since I sail in an area of the Pacific that often puts vessels in peril, I personally consider keeping the radio on and tuned to channel 16 to be a courteous practice and in keeping with the traditional "law of the sea," i.e., the ancient maritime practice of providing assistance to fellow seafarers (which is also the law: 46 U.S. Code § 2304.). But as skipper, you may disagree and prefer to conserve your batteries for your own vessel's emergency use.