What is the theoretical minimum size of solar cells on wrist watches?
Modern Silicon solar cells (PV cells deliver typically 15%+ module efficiency and about 20% is available using the high efficiency (and harder to use) Sunpower cells.

At one sun insolation = 100,000 lux output is

  1000 x A x Z Watts


A = Area in m^2, Z = efficiency (0 &lt;= Z &lt; 1)

For A in mm^2 above formula gives micro Watts.
So in full sun at say 15% efficiency you get  150 uW / mm^2 in full sun.

Bright sky on a sunny day or bright cloud on an overcast day with sun not visible but eye hurting bright is maybe 10% of full sun. Shadow with no direct line to sun is much lower - maybe 0.1% - 1%
Full sun is 100,000 lux so 1% = 1000 lux and 0.1% is 100 lux.  

Bright monitor screen backlight full white with cell against surface is about 300 lux = ~~~ 0.3% of full output.
Very bright room - 100 to 1000 lux.
100 lux - view colour pictures with ease (despite what text books say).
20 lux - view colour OK once eyes accustomed. Brighter would be nicer.

Watch covered by shirt sleeve etc ~~= 0lux.
Wear watch at angle to light main source = 10% of available if lucky.  

SO a watch in artificial light conditions MAY see 100 - 1000 lux range and less sometimes.

1000 lux = 1% of full sun.
So PV panel gets ~~ 1.5 uW mm^2 at 1000 lux.
Or 0.15 uW/mm^2 at 100 lux.
So your 1 uW watch meeds about 6 mm^2 if illuminated at 100 lux always.
If you can manage 10mm x 10mm = 100 mm^2 then you need about 1/16th of the time at 100 lux or about 1.5 hours per day. A brief burst of full daylight makes a major difference.

E&amp;OE.

ie about 30mm x 30mm or over 1 square inch.

100 lux = 10% of 1000 lux output.