Scientists at Massachusetts Institute of Technology noticed how the spiders' secretion absorbed water, helping to secure their next meal.
The sticky tape does the same and was found to work within seconds in tests on pig skin and lungs.
The team said with more research, it could be used in place of sutures.
But they are still several years away from trials in humans.
Getting tissues in the body to form a tight seal is difficult because water on their surface makes them slippery.
Sutures - stitches that hold a wound or cut together - don't always work well and can cause infections and pain.
And tissue glues, which already exist, can take several minutes to work and may drip on to other body parts.
So the scientists turned to nature for inspiration.
Spiders secrete a sticky material containing charged polysaccharides that absorb water from the surface of an insect almost instantaneously, leaving a small dry patch the glue can then stick to.
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