Fritz's second-round clash with rising Brazilian star Fonseca was suspended at one set all due to bad light on Wednesday.
Defending champion Fritz took the first set 6-3, but the 18-year-old Fonseca, ranked 57th, hit back to win the second 7-6(5).
When play resumed on Thursday morning, Fritz won the deciding set 7-5 to reach the last eight.
The 27-year-old, who won the grass-court tournament in Stuttgart earlier in June, returned to court just hours later and was pushed to the brink by fellow American Giron.
Top-seeded Fritz eventually prevailed against the world number 46, winning 7-5, 4-6, 7-5 to book a semi-final date with Spain's Alejandro Davidovich Fokina on Friday.
Davidovich Fokina, ranked 28th, beat Czech third seed Jakub Mensik 6-4, 7-5 in the quarter-finals.
The other semi-final pits French fourth seed Ugo Humbert against American world number 149 Jenson Brooksby.
Humbert beat Billy Harris 7-6(4), 6-1, while Brooksby, who entered the tournament as a 'lucky loser' after a withdrawal, defeated Dan Evans 6-2, 6-3.