Can mobs spawn on string?

No, string will not stop mobs from spawning - due to its use in redstone mechanisms as a means of detection.

String can be placed as tripwire, which can be used to detect entities as part of a tripwire circuit. Observers can also be used with a tripwire to generate Redstone signals.

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This makes it useful to detect when mobs spawn, to trigger traps, etc.

What it does block is natural generation of "growth", such as if it is placed on bamboo or sugarcane, it will no longer grow, and it will also stop the (natural) placement of snow layers in snowy biomes.

What you could use instead, is carpet. You could use a similar colour to your block palette, making it barely noticeable.

Carpet has the same thickness as a pressure plate. It has a hitbox of 1⁄16 of a block and covers the entire surface of the block it is placed on, unlike pressure plates. This thickness prevents mob spawning.

Source.