Can the piercing scream of a small child rupture an adult's eardrum?

No - although it may sometimes feel like your eardrums are on the threshold of rupturing...

It requires 178 dB (decibels) minimum to rupture a healthy eardrum (usually more). A baby's cry is around 115 -130 decibels at a distance of ~5" (about 10 cm) from the baby's mouth so it may be painful but it won't rupture your eardrum.

Note that a sustained level of about 95 dB can eventually lead to some hearing loss - but most babies can't sustain their crying long enough to cause that kind of damage.

On the other hand 125 dB is the threshold where sound starts getting painful - a loud baby can definitely reach that threshold.

Permanent hearing loss from a single exposure or maybe just short term exposure take about 140 dB - which no baby is going to generate (dB is a logarithmic scale).