I checked
star alliance web site and found
United Star Alliance Gold customers may only access the United Clubs within the U.S. when travelling in conjunction with a Star Alliance international flight.
Judging from this, TK*G can gain access to United Club whether international or not.
According to
UA web site. I'm confusing about the meaning of "OR".
A valid Star Alliance Gold membership card and a same-day boarding pass for travel on a flight operated by a Star Alliance member airline and departing from the same airport
OR
A same-day boarding pass showing Star Alliance Gold status, for travel within 24 hours on a flight operated by a Star Alliance member airline
I'm wondering there's no simple and
single rule for non-UA *G lounge access.
As airoli mentioned above,
star alliance reference guide make me confused too with the word "MAY".
UA and SAG customers may only access the UA United Clubs within the United States when travelling in conjunction with a Star Alliance international flight
This topic is discussed on UA Forum
refused PHX United Club access with a TK gold card. You can find many refused cases.