This is highly disputable. To the extent that it was true during the Cold War, it certainly isn't true post 9/11. The middle east is neither stable nor secure, and Russia has managed to start a deniable proxy war with the west by destabilising Ukraine and Syria resulting in a huge number of refugees.
Meanwhile US anti-drug and anti-communist efforts have left much of south America plagued with violence and recovering from dictatorships.
The thing that is holding the world together is cross-border trade. Both sides in the Ukraine are dependent on keeping the pipeline open, so they've not blown it up. China depends on exporting to the US. I'm not too worried about the Russians nuking London because half of it belongs to Russian oligarchs.
Stability as absence of conflict is a utopian measure. Stability as the truncation, resolution and containment of confict is a realistic measure attribuatable largely to the United States. Recent events and the US economic situation suggest the US cannot continue to play the same constructive role, and as such I fear greater instability is a certainty.
Tell me more about that constructive role in the Honduras coup as late as 2009.
The scorecard for "truncation, resolution and containment" really does not look great. The unambiguous successes are probably just Yugoslavia and the UK's intervention in Sierra Leone.
Maybes: Kashmir, Afghanistan, Sudan?
Ongoing failures: Ukraine, Yemen, Iraq, Syria, South America ("war on drugs") esp Mexico and Colombia, Libya, etc
Britain was beaten hard in the two wars.
EDIT: To see how the U.S.-U.K. alliance was not necessarily the sure thing it appears to be in hindsight, look into War Plan Red (https://en.wikipedia.org/wiki/War_Plan_Red). The U.S. actively maintained a strategic plan for fighting a war with Britain as late as 1939.
It is absolutely a 0sum game for China.
These are philosophical discussions and large depend on your own perspective and bias in judging the outcomes.
My personal take is that some evidence exists in that last global open war with direct fighting on all fronts being 70+ years ago (WWII) the world is doing something right.
Edits to fix spelling.