File format not recognized; treating as linker script using GCC

You have two problems with your gcc command-line. First, you’re specifying the -S flag, which causes gcc to emit assembly code, rather than object code. Second, you’re missing the -c flag, which tells gcc to compile the file to an object file, but not link it. If you just remove -S and change nothing else, you’ll end up with an executable program named file1.o and another named file2.o, rather than two object files.

Besides those errors, you could simplify your makefile by the use of pattern rules. I suggest you try the following instead:

all: exFile
exFile: file1.o file2.o 
    gcc -Wall -g -m32 $^ -o $@

%.o: %.c
    gcc -Wall -g -m32 -c $< -o $@

file1.o: file1.h

Or, as EmployedRussian points out, you can go with something even more minimal that leverages more of the built-in features of GNU make:

CC=gcc
CFLAGS=-Wall -g -m32

all: exFile

exFile: file1.o file2.o 
        $(LINK.c) $^ -o $@

file1.o: file1.h