This means British skaters have still not won an event at the Europeans since Dame Jayne Torvill and Sir Christopher Dean in the ice dance in 1994.
Fear, 26, and Gibson, 31, had a great chance to end that run, after they were in second following the rhythm dance, 1.5 points behind Fournier Beaudry and Cizeron.
But after a strong start where they performed their most difficult lift perfectly, Fear slipped slightly into the synchronised twizzles, the second element of the routine.
And while they got back on track and peaked brilliantly at the conclusion as bagpipes blared around Sheffield, their looks of disappointment on the ice at the conclusion were telling.
Instead Fournier Beaudry, 33, and 31-year-old Cizeron – who only teamed up in March – seal gold.
They became a pair after Cizeron had an acrimonious split with former partner Gabriella Papadakis, who has accused him of controlling behaviour.
Meanwhile, Fabbri, 37, and 36-year-old Guignard are the oldest male and female skaters at these championships and returned to Sheffield where they competed in their first European Championship together in 2012.
This was Fabbri’s 14th Euros, a new record – and he earned a sixth medal.
Earlier, fellow Brits Phebe Bekker and James Hernandez performed a smooth, spine-tingling routine that drew a standing ovation in Sheffield.
After starting well, the young duo – aged 20 and 24 respectively – only got better, with their choreographed spin and step movement towards the conclusion of the routine scoring well above their base value in the grade of execution.
It earned them a season-best 106.75 in the free dance, which put them in first place overall at the half-way point.
They finished 11th but have ensured they will be worth watching in years to come – and first, at the upcoming Winter Olympics.