In theory, yes - but there are historical in-universe explanations to caveat it. the general story that's told is that the way in which deities take power and channel it to their faithful creates a substantial difference - enough that a mortal making intellect-based constructs to draw the weave out directly just don't achieve. However, this story HAS changed over various game generations, and the reason behind it is at the pure mechanical balance and style level of the game itself; Arcane casters don't get access to healing magic, as a general principle of game design in D&D. Similar to the general rule that cantrips can never provide actual healing - you'll note that there are none that do, and that's absolutely deliberate.
Jump to the modern game rule-set, however, and some of these things are relaxing slightly, or they are being suggested at relaxing. We now have Divine Soul Sorcerer, for example - this is a sorcerous origin that, as their primary perk, lets you choose spells from the Cleric spell list as well as the sorcerer one. It's a major choice though - it's the cost-resource of picking that specific thing as you origin, to the exclusion of all else.
Similarly, there was an initial trial stage in the unearthed arcana for a Theurgist Wizard school - this would have been a wizard tradition that gave you access to cleric spells; once again it would have been a severe investment - the choice of your arcane tradition. That subclass didn't get enough traction and had some substantial balance problem though, and it didn't make it out of unearthed arcana. So, in theroy, yes, it's possible - but a wizard would need to be dedicating the whole of their wizarding career and study to learning how to manipulate the weave in ways that mimic divine power; a major, major undertaking that defines their entire academic life... not an afternoon looking at a cleric scroll you found in a shop.
The spell list explanation is mechanical - it's just simply an explanation of the rules of the game and the way it's set up, and the spell lists are set up the way they are for a good number of deliberate reasons. There are a number of spells that only wizards can learn, and which don't exist on any other class list. There are also a lot of spells that do not appear on the wizard list - traditionally divine ones and ones that are directly manifestations of divine connections. The reasons for this, at the base level, underneath the lore explanations and excuses, are balance-mechanical.
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As for the gods.... Gods are not all incarnations of law or order. Some are... but gods come in every possible flavour you can conceive of, more or less. There are gods for everything, and there are as many evil and neutral ones as goodly ones, and as many chaotic ones as lawful ones.... and a lot of gods have a very long history of an almost Grecian-deity style of pettiness and internal rivalry at times.
Gods in the realms hold what are referred to as folios (this is just the nomenclature that we use); specific things that they preside over and oversee. There is, for example, the folio of death, which is different from the folio of dying, and different from the folio of the transition between life and death, and different again from the folio of the dead, and not to be confused with the folio of murder, of assassination, of accidental death, of decay, of undeath.... all differentiated *enough* that different deities can hold them and preside over them... you may start to get the picture ^.^ The larger and more broad a folio is, the more powerful the god that controls it is - and must be to keep it. Begin a god is a dangerous business, because control of divine folios changes hands far more frequently than your average mortal might suspect, and a god that loses control of a folio to another is greatly diminished by the loss. Worse, a god can be subsumed by another having its folio absorbed into those controlled by the dominant god, and they risk being utterly destroyed in the process... and this, in turn, can happen as a result of mortal beliefs, souls and prayers, because in the realms, having belief and believes is a two-way street, and deities do gain - and lose - power by the strength and count of the faith of their followers.
Gods are a complicated topic, to put it simply ^.^
Oh! And, a number of today's gods were originally mortals who ascended to god-hood through various means, and most gods that exist have been forced at one time or another to be fully mortal too - and many were killed while they were, which caused a lot of problems.
If you'd like some more in-depth reading, you can try these pages:
https://forgottenrealms.fandom.com/wiki/Deityhttps://forgottenrealms.fandom.com/wiki/Portfoliohttps://forgottenrealms.fandom.com/wiki/Portal:Deities
