Felipe also called for a Brexit deal that would provide "stable living conditions" and "certainty" for hundreds of thousands of Spanish nationals living in Britain and Britons living in Spain.
Watch the entire speech by the 49-year-old Spanish king, delivered in impeccable English:
During the speech he paid tribute to those killed in recent terrorist acts, including Labour MP Joe Cox who was brutally murdered a year ago, PC Keith Palmer, the policeman killed in the Westminster Bridge attack and more recently, Spaniard Ignacio Echeverria, the skate-boarding hero who was killed in the London Bridge attack.
But rather than risk stirring tensions with a direct call to reclaim the sovereignty of Gibraltar, he spoke in a concillatory tone, expressing his confidence the two nations could overcome their "differences" over Gibraltar and find a solution "acceptable to all involved".
He said: "It is just as true, however, that during our rich and fruitful history there have also been estrangements, rivalries and disputes, but the work and determination of our governments, authorities and citizens have relegated such events to the past.
"I am certain that this resolve to overcome our differences will be even greater in the case of Gibraltar and I am confident that through the necessary dialogue and effort, our two governments will be able to work towards arrangements that are acceptable to all involved."
He also couldn't avoid mention of Brexit, admitting that the situation "saddens" Spain but that it fully respected the result of last year's vote.
Felipe spoke about the hundreds of thousands of Spaniards living in Britain and Britons living in Spain, whose future hangs in the balance as Britain negotiates its withdrawal from the EU.
"These citizens have a legitimate expectation of decent and stable living conditions for themselves and for their families," he said, calling for a deal that would provide "sufficient assurance" to them.
There are an estimated 300,000 British citizens living in Spain -- the majority of them retirees -- and around 116,000 Spaniards living in Britain.
Britain's Prime Minister Theresa May opposition Labour party Leader Jeremy Corbyn ahead of a speech by Spanish King Felipe VI at the Palace of Westminster
- In pics: Spanish royals on parade in state visit to London
- Spain's King and Queen start state visit to UK amid Brexit tensions