I reviewed quickly the Adafruit ST7735 library and noticed that it uses a shield with an SD, so the examples will need a couple modifications to use the SD of the Galileo, but the only avr library that I saw is the avr/pgmspace.h, you can take a look at this post https://communities.intel.com/thread/46076 , in there is a link to a library provided by Mikal.Hart that should let the IDE compile the code that uses the #include <avr/pgmspace.h>. You can give it a try to that library and the workaround it might work for you.
This does not really help. In most libraries I tried up to now after replacing this, only the next and over next dependency to avr header files follows.
I have not seen any ST7735 library alternatives that do not depend on AVR. Any library that refers to AVR will not work. If by modifying the code allows the sketch to compile, there is a big chance it will not run properly. The reason being that AVR refer to registers found on the Atmel chip on Arduino Boards and Galileo has a Quark x86 processor.