Not as intended.
A 7speed shifter will move the cable 5mm per click. 6 speed sprockets are 5.5mm apart.
So if you set the cable tension so it's correct in the middle of the cassette/freewheel, it will be 1.5mm short of reaching the correct position for the top or bottom gear sprockets. That will be too far off to work.
The only entry on CJ's tables that would work is a 7 speed Shimano shifter with an old Dura-ace rear mech. That's no good as an old Dura-ace mech will be just as hard to find as a 6 speed shifter, and the road mech won't like large sprockets.
Probably the best bet is to use a Jtek model 2
Shiftmate from
SJSC.
This is sold for using a 9 or 10 speed shifter with an 8 or 9 speed cassette (or vice versa). According to CJ's table the ratio between 6 and 7 speed cable pulls is much the same as that between 8 and 9 or between 9 and 10, so it should work.