No. The default setting is the multiverse of all D&D worlds, which includes the Forgotten Realms.
Jeremy Crawford, in 2015, states this in a tweet:
Does the #dnd tabletop RPG have one official setting? The answer is yes. That setting is the multiverse, which includes all #dnd worlds.
In the reply thread to that tweet, he directly answers this question:
The core books intentionally emphasize the multiverse. FR is the focus of OP and our first #dnd5e adventures.
In other words, Forgotten Realms is not the default setting, not in the way that Greyhawk was the implied setting of D&D third edition, and we see this in the core rulebooks where the deities are not limited to those of one setting, but mention Greyhawk, Forgotten Realms, Eberron and pantheons of real-world myth.
Jeremy Crawford describes in his twitter thread how this multiverse works in spite of seemingly incompatible cosmologies:
The worlds occupy pockets of the Material Plane—sort of like planets but in a space shaped by magic and divine forces.
The inhabitants of each world in the Material Plane have different ways of imagining the multiverse (check out chapter 2 in the DMG).
In a 2013 article, James Wyatt described this multiverse approach as a deliberate design choice in D&D 5th edition (called in this article by its working title D&D Next), so we know it's not just Crawford coming up with this on the spot:
Pretty early on, we agreed that the core rules for D&D Next had to acknowledge the existence of all the worlds of D&D—not just the Forgotten Realms we’ve been talking a lot about, but also Greyhawk, Eberron, Krynn, Athas, Mystara, Ravenloft—and, most importantly, the thousands and thousands of worlds created by DMs for their own games.
Wizards of the Coast has decided to make Forgotten Realms the default setting of several of their published adventures. For example, Princes of the Apocalypse is set in Forgotten Realms, with a conversion guide for other settings. Tomb of Annihilation is similarly set in Chult, and Storm King's Thunder is explicitly set in the Realms too, as is Lost Mine of Phandelver.
On the other hand, Tales of the Yawning Portal contains three adventures which have no default setting, three set in Greyhawk, and one in Forgotten Realms by default. Out of the Abyss is set in the Underdark, a Realms feature of Greyhawk origin, and Curse of Strahd is set in Ravenloft. Mordenkainen's Tome of Foes is named for a Greyhawk character.
In short, while D&D 5th edition material makes heavier use of Forgotten Realms than any other D&D world, it is not the default setting; officially, it's only part of the larger D&D multiverse that includes all D&D worlds.