Brief Answer:
No

Detailed Answer:
There is no natural remedy for hypothyroidism. Of course, this is true of correctly diagnosed hypothyroidism. Because, albeit uncommonly, there are temporary elevations of the TSH with or without low thyroid hormone levels that may normalize over the course of several weeks. This condition is called thyroiditis.

An endocrinologist can differentiate between these two conditions by performing a thorough assessment in person , in addition to appropriate blood tests such as
CBC
CMP
TSH
Free T4
Thyroid antibodies (anti TPO and anti Thyroglobulin)

Furthermore, follow up visit and testing may well be necessary in a couple months to see if the thyroid has normalized.

However, if the initial labs are categorically suggestive of Hypothyroidism (for example: very high TSH, low Free T4, positive thyroid antibodies) then this approach is not necessary because the implication of such a diagnosis is that this is an irreversible lifelong condition which requires medication for life.

And finally, this whole discussion applies to the most common form of hypothyroidism worldwide which is Primary Acquired Mild/Overt Antibody positive/negative Permanent hypothyroidism due to Chronic (Hashimoto's) Thyroiditis. These words simply mean that the thyroid gland is not working well enough to produce adequate amounts of thyroid hormone as it has been damaged irreversibly by autoimmunity ('body's own defenses).
It also applies to situations where the thyroid gland has been surgically removed or damaged by radiation.
This discussion does not apply to 'Secondary' hypothyroidism in which the hypothalamus or pituitary gland do not produce enough TSH to stimulate the thyroid gland. Sometimes Secondary hypothyroidism can well be temporary till the pituitary problem is resolved. However, many times Secondary hypothyroidism can also be permanent. In these cases too, levo thyroxine is the only medication that should be taken.
Expert thyroid guidelines have repeatedly emphasized this fact

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