GRAY Graybug Vision Inc

Exhibit 5.1

Carlos A. Ramirez

+1 858 550 6157

cramirez@cooley.com

March 29, 2024

CalciMedica, Inc.

505 Coast Boulevard South, Suite 307

La Jolla, California 92037

Ladies and Gentlemen:

We have acted as counsel to CalciMedica, Inc., a Delaware corporation (the “Company”), in connection with the offering by the Company of shares of its common stock, par value $0.0001 per share (“Common Stock”), having an aggregate offering price of up to $17,068,386 (the “Shares”) pursuant to the Registration Statement on Form S-3 (File No. 333-273949) (the “Registration Statement”) filed with the Securities and Exchange Commission (the “Commission”) under the Securities Act of 1933, as amended (the “Securities Act”), the prospectus included in the Registration Statement (the “Base Prospectus”) and the prospectus supplement dated March 29, 2024 filed with the Commission pursuant to Rule 424(b) promulgated under the Securities Act (together with the Base Prospectus, the “Prospectus”). The Shares are to be sold by the Company under that certain At the Market Offering Agreement, dated August 11, 2023, between the Company and H.C. Wainwright & Co., LLC (the “Sales Agreement”), as described in the Prospectus.

In connection with this opinion, we have examined and relied upon the Registration Statement, the Prospectus, the Sales Agreement, the Company’s certificate of incorporation and bylaws, each as currently in effect, and such other records, documents, certificates, opinions, memoranda and instruments as in our judgment are necessary or appropriate to enable us to render the opinion expressed below. As to certain factual matters, we have relied upon a certificate of an officer of the Company and have not independently verified such matters.

In rendering this opinion, we have assumed the genuineness of all signatures; the authenticity of all documents submitted to us as originals; the conformity to originals of all documents submitted to us as copies; the accuracy, completeness and authenticity of certificates of public officials; and the due authorization, execution and delivery of all documents where authorization, execution and delivery are prerequisites to the effectiveness of such documents.

We have assumed (i) that each sale of the Shares will be duly authorized by the Board of Directors of the Company, a duly authorized committee thereof or a person or body pursuant to an authorization granted in accordance with Section 152 of the General Corporation Law of the State of Delaware (the “DGCL”) and (ii) that no more than 17,068,386 Shares will be sold for a consideration not less than the par value of the Common Stock. We express no opinion to the extent that future issuances of securities of the Company and/or anti-dilution adjustments to outstanding securities of the Company cause the number of shares of Common Stock available for issuance by the Company to be less than the number of Shares then issuable under the Sales Agreement.

Cooley LLP 10265 Science Center Drive San Diego, CA 92121

t: (858) 550-6000 f: (858) 550-6420 cooley.com

CalciMedica, Inc.

March 29, 2024

Page Two

Our opinion herein is expressed solely with respect to the DGCL. Our opinion is based on these laws as in effect on the date hereof. We express no opinion to the extent that any other laws are applicable to the subject matter hereof and express no opinion and provide no assurance as to compliance with any federal or state securities law, rule or regulation.

On the basis of the foregoing and in reliance thereon, and subject to the qualifications herein stated, we are of the opinion that the Shares, when issued and paid for in accordance with the Sales Agreement and as provided in the Registration Statement and the Prospectus, will be validly issued, fully paid and nonassessable.

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We hereby consent to the reference to our firm under the caption “Legal Matters” in the Prospectus and to the filing of this opinion as an exhibit in the Current Report on Form 8-K to be filed with the Commission for incorporation by reference into the Registration Statement. In giving such consent, we do not thereby admit that we are in the category of persons whose consent is required under Section 7 of the Securities Act or the rules and regulations of the Commission thereunder.

Our opinion set forth above is limited to the matters expressly set forth in this letter, and no opinion should be implied or may be inferred beyond the matters expressly stated. This opinion speaks only as to law and facts in effect or existing as of the date hereof, and we have no obligation or responsibility to update or supplement this opinion to reflect any facts or circumstances that may hereafter come to our attention or any changes in law that may hereafter occur.

Cooley LLP 10265 Science Center Drive San Diego, CA 92121

t: (858) 550-6000 f: (858) 550-6420 cooley.com