In as much as the Royal Navy was the Monroe Doctrine, yes...
Very much so. When President Monroe announced that the hemisphere was closed, the US did not have the capacity to enforce it, but the British could. In return Britain was able to extend the reach of its informal empire.
To a degree, it could be argued that in the absense of a official American doctrine, the British would have enforced something similar, though probable the Dutch, French and even Spanish would have had more leeway.
Something which might have changed is Portuguese-Brazil relations given that the Doctrine came in less than a year after independence and the Brazilian Emperor was also the heir presumptive of Portugal, but it’s likely a similar outcome would’ve occured.