A spiritual weapon (created by the spell of the same name) uses your bonus action for movement and attacking. It does not have its own actions, bonus actions or reactions.
The Spiritual Weapon spell has a casting time of 1 bonus action and range of 60 feet. Breaking down the spell description:
You create a floating, spectral weapon within range ...
So, you use your cleric's bonus action to cast the spell, creating the spiritual weapon within 60 feet of your cleric.
When you cast the spell, you can make a melee spell attack against a creature within 5 feet of the weapon.
If a creature is within 5 feet of where you created the spiritual weapon, as part of the same bonus action you used to cast the spell, you can attack that creature with the spiritual weapon.
As a bonus action on your turn, you can move the weapon up to 20 feet and repeat the attack against a creature within 5 feet of it.
In every round after you cast the spell, for its duration, you can use your bonus action to move the spiritual weapon up to 20 feet and attack a creature within 5 feet of the spiritual weapon.
Since any use of the spiritual weapon spends your bonus action, your cleric still has a regular action (and reaction) available.
Since it doesn't specify (beyond the fact that the weapon floats) it's up to DM discretion, but I expect most DMs would allow it (I would, mostly because it moves slowly enough that it's unlikely to matter much against most flying enemies).
That said, the Flaming Sphere spell does have specific rules about how large a gap or high a barrier it can jump, so it wouldn't be outside the bounds of possibility for a DM to rule that the Spiritual Weapon can't do that sort of thing because the spell isn't explicit about it.
Best Answer
No; the spiritual weapon is used by the caster
The text in the spell description is pretty clear on who makes the weapon attack: you (the spell caster). It is also clear on who causes it to move.
What happens when the spell is cast is that it creates a magical effect. If someone else were able to use this weapon as you suggest in your question, they would be making a melee weapon attack, or a melee attack, not a melee spell attack. (Melee attack, PHB p. 195; spell attack rolls, PHB. p. 205).
Why is it important that this spell specifies a melee spell attack? Attacks that are ranged (as opposed to melee) have disadvantage on when opponents are within 5 feet of you. (PHB. p. 195 & p. 205).
As another clue to the spell's function, the spell is described as being from the evocation school, rather than conjuration school. Were a weapon to be created or summoned by this spell, it would most likely have been classified as conjuration.
Spells in this edition generally do what is in the spell text. If this magical effect created a weapon that was transferable to another creature, that would be mentioned in the spell's text.
Compare this to a different spell, like Goodberry
In this case, a creature indicates that use of the berries is not restricted to whomever cast the spell.
For another example, see Magic Stone
If the official rulings from J. Crawford are of interest, he says this: