My oldest had a blocked tear duct for nearly a year (they usually clear on their own by about 6 months). We got used to the goopy eye look. I would clean her eye with a clean cloth each morning or use breast milk to get the really bad goop off. I attempted to massage the corner of her eye as suggested by my midwife and doctor, but it never helped. If it had gone past a year we were looking at intervention to open it up, but it finally went away on it's own (their faces grow and the duct opens as it has more room). Incidentally neither of my girls got the eye drops at birth but only one got a plugged duct, it's just random.
Now, all that being said if baby's eye is red or puffy and gooping out green stuff, that would warrent a doctor's visit, IMO.
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